What happened to Ser Arthur Dayne's sword Dawn after Ned brought it back to Ashara?

GRRM: Dawn remains at Starfall, until another Sword of the Morning shall arise. -SSM, Valyria-related Subjects: 1 Jan 2002

Dawn in TWOW

Note: I am going to try and provide a little background on the Daynes/Dawn here quickly, but its been done much better by others. If you have a good understanding of House Dayen and the sword Dawn, just skip down to the "Dawn coming into the Storyline" section

Background

House Dayne can trace their ancestry back to the "dawn of days", but it should be noted that they are not necessarily Valyrian and are more likely descendants of the First Men (do note that GRRM didn't say they weren't valyrian):

GRRM: I would have to consult my notes to tell you the Dayne arms. Offhand I don't recall. As for the violet eyes . . . look, Elizabeth Taylor has violet eyes, and she's not of Valyrian descent (that I know). Nor is she related to Aegon the Conquerer. Many Swedes have blue eyes, but not all those with blue eyes are Swedes, and not all Swedes have blue eyes. The same confusions exist in the 7 Kingdoms.

Ran: So many banners, I was shooting in the dark. And I was actually arguing that she was a Liz Taylor type. Thanks. :)

George_RR_Martin: If you want to figure out a family's descent, the names are a better clue than the eyes. Stark. Reed. Flint. Tallhart (tall hart). SSM, Event Horizon Chat: 18 March 1999If interested: The Dayne Name

So this super secret First Men (but valyrian looking) house comes across a fallen star and not only forge a sword out of it, but also name their castle after it (Starfall).

Dawn is like no sword that has ever been seen before:

"The finest knight I ever saw was Ser Arthur Dayne, who fought with a blade called Dawn, forged from the heart of a fallen star. -ACOK, Bran III

and:

At the mouth of the Torrentine, House Dayne raised its castle on an island where that roaring, tumultuous river broadens to meet the sea. Legend says the first Dayne was led to the site when he followed the track of a falling star and there found a stone of magical powers. His descendants ruled over the western mountains for centuries thereafter as Kings of the Torrentine and Lords of Starfall. -TWOIAF: Dorne: Kingdoms of the First Men

and:

The Daynes of Starfall are one of the most ancient houses in the Seven Kingdoms, though their fame largely rests on their ancestral sword, called Dawn, and the men who wielded it. Its origins are lost to legend, but it seems likely that the Daynes have carried it for thousands of years. Those who have had the honor of examining it say it looks like no Valyrian steel they know, being pale as milkglass but in all other respects it seems to share the properties of Valyrian blades, being incredibly strong and sharp. -TWOIAF, Dorne: The Andals Arrive

I've previously compared it to not only the blades of the Others, but also their bones. Comparing Dawn & the Bones of the Others

But moving along, lets get to how does this sword come into play in the series?

Dawn coming into the Storyline

In a parallel universe where GRRM still has a 5 year gap in play it seemed very possible that in addition other children/young adults developing their craft (Bran - magic, Arya - assassin, Sam - citadel, etc.) that GRRM intended to have young Edric "Ned" Dayne become a Sword of the Morning.

With that gap not existing here, we are stuck with an 12-13 year old Edric Dayne, that for some reason, no one in Dorne seems to care about.

Gerold Dayne

Gerold Dayne aka Darkstar is from a cadet branch of House Dayne that is located at High Hermitage. Gerold's attempt on Myrcella's life changed a ton in Dorne and now Obara (snake), Balon Swann (kingsguard) and Areo Hotah (POV) are suppposedly heading to catch Darkstar in his den:

The girl will ask him to hunt down the man who hurt her. If he is the man I judge, Swann will not be able to refuse. Obara, you will lead him to High Hermitage to beard Darkstar in his den. -ADWD, The Watcher Edric Dayne

If interested: The Showdown at High Hermitage

Bu it should be noted that High Hermitage is just northeast of Starfall on the Torrentine river. It makes more sense (at least for me) and matches Darkstar's character that he could attempt to steal Dawn (not a new theory) and become a default Sword of the Evening (wielders who use Dawn through tradional means are known as Sword of the Morning, we only know of one other Sword of the Evening, and it could have been a name just for him (Vorian) but with Darkstar being "of the night" and wielding a sword that was not meant for him/killing a member of House Dayne could make him the "Sword of the Evening".

Darkstar is "the most dangerous man in Dorne" and I don't doubt that Hotah could probably beat him with equal weaponry, but if Darkstar has Dawn he obviously gains an edge.

After that is where speculation gets hard, as Darkstar's attempt on Myrcella rules out his allegiance there, if the Dornish side with Young Griff that would also rule him out probably as well. That said I do think a Gerold Dayne stealing Dawn is our best bet for how the sword comes back in.

Edric "Ned" Dayne

Ned Dayne (Beric Dondarion's squire in the BWB) and several other BwB members apparently splinter off from the brotherhood once Lady Stoneheart takes command:

"We were king's men when we began," the man told her, "but king's men must have a king, and we have none. We were brothers too, but now our brotherhood is broken. I do not know who we are, if truth be told, nor where we might be going. I only know the road is dark. The fires have not shown me what lies at its end." -AFFC, Brienne VIII

and we see in the AFFC, Appendix a separation of groups. While some people think that they may save the day for Jaime/Brienne or have infltrated the Vale, it remains possible that Ned (and other missing members of the BwB) such as Anguy (Dornish winner of the archery contest in the Hands Tourney), etc.

As I mentioned above, Ned seems a bit young to be declared a Sword of the Morning, but he has been missing since ASOS (to the reader) and AGOT (to the Dornish, who don't seem to care lol).

A Combination of the Two

Another thought I had was that they both could show up at Starfall. Darkstar with his supporters from High Hermitage, Ned with the missing BwB members and then Obara/Hotah/Balon and there is a fight over the sword, in which we get a ton of background on House Dayne, etc.

Someone Else

Obviously someone not of House Dayne could (and probably will) wield it (see Elio/Linda's comments below), but that will probably be later in the story (Ned giving it to Jon, Darkstar swearing it/or himself/both to Young Griff, etc.). I really think that due to proximity, POV, etc. it has to be one of Darkstar/Ned to bring it into the storyline (but not wield it the entire time). That said in the unlikely case that it isn't one of the two Daynes (Edric/Darkstar) it does remain possible that someone in one of their groups (or Hotah/Obara/Balon) could wield it for a short period of time.

TWOW & Beyond

GRRM has stated that in order to be named Sword of the Morning:

the Sword of the Morning is always a member of House Dayne, someone who is deemed worthy of wielding Dawn as decided within the House, that whoever it is would have to earn the right to wield it.

So that is why I agree with regards to what exact Dawn is and how it will be used, I will note this comment by Elio/Linda (this is their personal belief, not something GRRM told them) that:

We think Dawn, the sword of House Dayne, was originally Azor Ahai's Lightbringer, and that the Sword of the Morning is an office created to carry and preserve the sword until Azor Ahai was reborn (we may actually have originated this one, way back in the dusty days of the EEsite forum).

TLDR: A look at the different ways that the sword Dawn could enter our storyline in TWOW.