Many portable CD players now play mp3 files on CD-R and CD-RW discs now. This could be a real option for an 11 year old and has the dual-use angle. The drawback is higher battery demand and larger size, but you can fit 700MB of files on a single CD-R. These players are pretty reasonable in price right now.
Hard disk based players are very pricey, I would have misgivings about a youngster carrying around a $400 music player everywhere, and potentially getting lost. They do have the advantage of being able to hold an entire very large music collection.
Flash-memory based players range from $50 to $300 based on how much memory capacity is on board. I've seen some $50 players at the local Wal-Mart with 128MB of memory. That is around 2 hours of music capacity if the files are encoded at 160Kbps or less. They also have the least battery drain, and typically very small size. Better ones have a built-in FM radio tuner also.
My preference if for flash memory players. I just purchased a player (my xmas gift to myself this year) that has 512MB memory capacity, built in FM tuner, recording capabilities, and a built-in FM transmitter for playing music to a radio without using an adapter. It is a $200 unit I sniped on ebay for $170.
Muro MR-100 mp3 player