Something I've done in my free time: I've come up with signature moves for all Gen 1 mons that count as fully evolved as of Gen 8.

For most cases I've given them powerful moves with effects (or combinations of effects) unique to them. In others they're just variations of existing moves (Golduck's is essentially a Water-type Psyshock) It's likely that a few of them are not balanced at all, but like I said, it's just something that I came up with since I love the concept of signature moves. To the extent I could I've based the moves around their appearance and/or dex entries (though there are some exceptions such as Omastar, since its dex entries are all either "died because shell was too heavy or "it eats lol"). In the descriptions I haven't specified whether the moves are physical or special, but they'll always correspond to whatever attacking stat is higher in that specific Pokémon (or in the case of Pidgeot and Mew, special). Tell me what you guys think about these!

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Venusaur:

-Bloom Blast: Grass type, 80 power, 100% acc, 20% chance of sleeping

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Charizard:

-Dragonfire: Fire type, 85 power, 100% acc, 15% chance of burning, super effective against Dragon

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Blastoise:

-Shell Cannons: Water type, 95 power, 100% acc, summons rain, 10% chance of flinching

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Butterfree:

-Noxious Powder: Bug type, 90% acc, paralyzes opposing Pokémon on the field, next opposing Pokémon to come out has its Speed lowered by one stage

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Beedrill:

-Poison Spear: Poison type, 85 power, 100% acc, 10% chance of poisoning, destroys Reflect and Light Screen, causes Protect to only reduce damage dealt by 50%

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Pidgeot:

-Mighty Gust: Flying type, 90 power, 95% acc, blows away screens and entry hazards

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Raticate:

-Ultra Fang: Normal type, 80 power, 95% acc, super effective against Steel and Rock, can hit Ghost types

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Fearow:

-Spear Dive: Flying type, 90 power, 90% acc, two-turn attack: first turn flies into the air and raises attack, second turn attacks

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Arbok:

-Bone Crush: Poison type, 25 power, 85% acc, traps target and inflicts 1/10 HP as damage for 2 turns, lowering the target's speed and defense by one stage each turn

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Raichu:

-Electro Tail: Electric type, 90 power, 100% acc, 15% chance of paralysis, raises attack by one stage

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Sandslash:

-Sharp Assault: Ground type, 70 power, 100% acc, +1 priority, fails if the opponent's move that turn isn't an attack, if it hits, the user will only take 2/3 from physical attacks that turn and the attacker will take 1/8 max health damage if said physical attack is a contact move

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Nidoqueen:

-Queen's Decree: Poison type, 100% accuracy +1 priority, lowers its target's two highest stats by one stage and raises those stats on itself by one stage. Can only be used on the first turn after coming onto the field

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Nidoking:

-King's Decree: Poison type, +1 priority, protects from opponent's next move. If an attack is used, the attacker takes 1/6 of its health as damage and the move that was used is disabled for the next three turns. Only one move may be disabled at a time. Fails if used succesively

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Clefable:

-Moon's Blessing: Fairy type, heals 50% max HP and resets all stat changes on the field

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Ninetales:

-Foxfire: Fire type, 60 power, 95% acc, curses target if it resists Fire

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Wigglytuff:

-Puff Up: Normal type, +1 priority, if attacked by a physical move, it will do no damage and the attacker will be switched out. If this occurs Wigglytuff will stay puffed up for 3 more turns, doubling its defense and special defense for the duration

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Vileplume:

-Pollen Cloud: Poison type, 80% accuracy, badly poisons opposing Pokémon and increases special attack and special defense by one stage for every target poisoned

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Parasect:

-Infection: Bug type, 85% accuracy, infects target, causing it to lose 1/8 HP every turn and have its attack decreased by one stage every turn, in turn raising Parasect's attack by one stage every turn

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Venomoth:

-Noxious Wave: Poison type, 80 power, 95% acc, 40% chance of either paralyzing or poisoning target

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Dugtrio:

-Tunnel Trap: Ground type, 70 power, 90% acc, the target falls into a hole, making it move last on its next move

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Persian:

-Quick Reflexes: Normal type, 80 power, 100% acc, uses speed stat instead of attack to calculate damage

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Golduck

-Hydrokinesis: Water type, 90 power, 100% acc, targets defense instead of special defense

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Primeape:

-Primal Rage: Fighting type, 120 power, 100% acc, is locked into the move for 2-3 turns and is then confused

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Arcanine:

-Blazing Majesty: Fire type, 85% accuracy, badly burns all targets hit, causing them to take 1/8 max HP damage every turn, otherwise acts like a regular burn

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Poliwrath:

-Breaststroke: Water type, 70 power, 100% accuracy, if it is raining, this move's power is doubled and has +1 priority

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Alakazam:

-Abracadabra: Psychic type, 65 power, 100% accuracy, power is increased by 50% if it targets an opponent with a held item, disables said item for the rest of the battle

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Machamp:

-Machamp Special: Fighting type, 100 power, 85% acc, first time any Pokémon is hit by this move it won't be able to move next turn (or simply after Machamp if it went first), accuracy drops to half after hitting a target

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Victreebel:

-Lure: Poison type, 75 power, 100% accuracy, only hits Pokémon that used/will use non-attacking moves on the turn it is selected, lowers defense by one stage

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Tentacruel:

-Poison Net: Poison type, 40 power, 85% accuracy, -1 priority, traps target for 2 turns and inflicts poison

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Golem:

-Crash Down: Rock type, 100% accuracy, power varies based on its weight compared to the target's, lowers defense by one stage

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Rapidash:

-Flaming Horn: Fire type, 60 power, 100% acc, is always a critical hit

(Galarian): Psycho Drill: Psychic type, 60 power, 100% acc, is always a critical hit

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Slowbro:

-Inspiration: Psychic type, increases Slowbro's Special Attack and Speed by three stages each for 2 turns, after which they drop back down 3 stages

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Dodrio:

-Triple Stab: Flying type, 30 power, 95% acc, hits 3 times, high critical strike chance

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Dewgong:

-Ice Horn: Ice type, 80 power, 90% acc, 30% chance of lowering defense and 15% chance of freezing

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Muk:

-Junkyard: Creates a field that increases the power of Poison type attacks by 20%, heals grounded Poison type Pokémon, poisons grounded non-poison immune Pokémon and turns Surf and Muddy Water into Poison type moves

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Cloyster:

-Hydro Spikes: Water type, 25 power, 100% acc, hits 2-5 times

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Gengar:

-Spook: Ghost type, 80 power, 100% acc, bypasses Substitute, lowers evasiveness by one stage

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Hypno:

-Sleep Control: Psychic type, 75% acc, puts target to sleep and can make it hit itself or an ally with a typeless 50 power special attack

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Kingler:

-Giga Grip: Water Type, 100 power, 100% acc, lowers speed by two stages

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Electrode:

-Electrobomb: Electric type, 250 power, 100% acc, faints upon using the move, creates Electric Terrain for 5 turns

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Hitmonlee:

-Spring Kick: Fighting type, 100 power, 95% acc, forcibly switches out target and switches user out as well

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Hitmonchan:

-Knockout: Fighting type, 65 power, 100% acc, deals double damage to targets below half health, increases Speed by one stage if it takes down a target

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Weezing:

-Miasma: Poison type, 90 power, 95% acc, 20% chance of confusing

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Kangaskhan

-Motherly Instinct: Normal type, 75 power, 100% acc, power doubles if below half HP

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Seaking:

-Hydro Drill: Water type, 85 power, 100% acc, 20% chance of confusing

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Starmie:

-Wonder Glow: Psychic type, 80 power, 90% acc, hits all opposing targets, 40% chance of lowering accuracy

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Mr.Mime:

-Master Wall: Psychic type, +4 priority, protects itself and all allies during the turn the move is used, reduces damage taken by 20% on the next turn, fails if used successively

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Jynx:

-Mesmerize: Psychic type, +3 priority, becomes the center of attention, reduces the accuracy of moves targeting it by 10%

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Pinsir:

-Pincer Grip: Bug type, 70 power, 100% acc, target cannot be switched out, 20% chance of lowering defense by one stage

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Tauros:

-Frenzy: Normal type, increases attack and speed by two levels each, but confuses itself and doubles damage received

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Gyarados:

-Hydro Rage: Water type, 120 power, user is locked onto this move for 2-3 turns and is then confused

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Lapras:

-Chilling Melody: Ice type, 80 power, 100% acc, damages all enemies, 10% chance of freezing, summons hail for 5 turns

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Vaporeon:

-Liquid Body: Water type, increases defense and special defense by one stage, if it is raining, increases special defense by one additional stage and heals 33% max hp

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Jolteon:

-Electro Needles: Electric type, 75 power, 100% acc, target takes 1/10 of its hp as damage after being struck by this move, as well as the turn after, with a 20% chance of being paralyzed on those turns

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Flareon:

-Nitro Boost: Fire type, 90 power, 100% acc, increases speed by one stage after hitting, 10% chance of burning

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Omastar:

-Depth Charge: Water type, 100 power, 90% acc, hits all enemy targets, confuses if weak to Water

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Kabutops:

-Aqua Slicer: Water type, 90 power, 100% acc, ignores a third of the target's defense

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Aerodactyl:

-Sky Prowess: Flying type, 75 power, 90% acc, avoids priority moves and hits foes that have used those moves for 20% increased damage

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Snorlax:

-Full-Body Crash: Deals damage based on the user's weight compared to the target's, chance of paralyzing scales similarly (15% min, 40% max)

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Articuno:

-Frost Zone: Creates a terrain for 5 turns that increases the damage of Hail by 50%, increases the power of Ice-type moves by 20%, prevents grounded Pokémon from being burned and decreases super effective damage taken by Ice-type Pokémon by 25%

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Zapdos:

-Thunderstorm: Electric type, 100 power, 85% acc, summons a thunderstorm for 4 turns, Pokémon that aren't immune to Electric take 1/3 damage at the end of the storm, increases the power of Electric-type moves by 50%

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Moltres:

-Firestorm: Fire type, 95 power, 95% acc, 20% chance of burning, scorches the target team's ground for 3 turns, damaging non-Fire type Pokémon for 1/10 HP every turn

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Dragonite:

-Marine Soul: Dragon type, 100 power, 100% acc, this attack strikes twice; a 70 power Dragon type attack and a 30 power Water type attack

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Mew:

-Adaptapulse: Psychic type, 75 power, 100% acc, always deals super effective damage