Minimal system for a small private home server

My oldest kids (8 and 10) asked if I could set up a server so that they would have a world available all the time (I apologize if my terminology is wrong, I don't play it myself). I have a home server that is up 24x7 and is used for file serving, streaming video to the media players and sharing the printer - if I were going to create a server for them, that's what it would run on.

I searched here for for a similar topic and turned up this thread, but it is a year old and I do not know if any of it still applies. I also Googled and read through the setting up a server tutorial and checked the can I host a Minecraft server tool. Both of these focus on bandwidth and memory as the deciding factor.

My media server is a fairly low end system: Atom D510 w/ 1GB RAM running Ubuntu Lucid Lynx (Server). I can order 4GB of RAM and have it in the system by the weekend so memory shouldn't be a limiting factor. The network is hard-wired gigabit ethernet so bandwidth shouldn't be a problem. However the CPU is a dual core 1.67GHz hyper-threaded "Pineview" Atom that is soldered to the motherboard. That one I cannot change.

I do not want to purchase 4GB of DDR2, that has absolutely no use outside of this system, unless there is a reasonable chance of this working. I'm not interested in running any mods - I'd set it up in "creative" mode and just let them build on it. I'm also not interested in pointing them to a random server on the internet instead.

Assuming that the server would see at most 4 simultaneous users, is this likely to be enough to run a Minecraft server?