It'd have to be SanDisk's Sansa Clip:
1GB, has a small screen but decent controls, and it uses a lithium-ion battery like said mother suggests. Normally goes for $40 but can be found for a bit less if you shop around. If you're in a real hurry, retail shops do carry these.
The main catches: it's Windows-only, and you can't use it with iTunes or songs you've bought from the iTunes Store (the Clip doesn't support AAC, protected or not).
I'm looking for some mp3 players to get some of my relatives for Christmas. Their mother told me that she had some specifics in mind:
Must be under or around $40
Must be 512 mb or 1 gb
Must have a display (she doesn't know much about mp3 players, I can't think of any that don't have a display of some sort)
Must not use batteries (i.e. you charge it like a phone or an iPod, rather than using AAA batteries)
Anyone have any suggestions?