simply put, phat is the old one
slim is newer than phat n divided into 2k n 3k series.

Old 2k can run custom firmware fine, while later one requires u to hv the power on always n if u wanna run custom firmware.

Custom firmware enables u to run homebrew n plugins that extends ur psp basic usage.

3k is the latest one before the PSP Go and it is pandora proofed.

IMHO, the old 2k version is the best but that one is hard to find nowadays if u're into tinkering the psp.

but since this is for ur nephew, n u wanna stick to simple n ease of use, just go to 3k version since it's the cheapest around n dun give much trouble if u just stick to playing via the legit ori umd.

New ori umd cost around 100 above depending on the titles while used ones u can get from 40 - 80 from the garage sales.