Gaming desktop for minecraft?

Anything pretty much, Mincraft has low requirements. Just pay about 600 for a new one and you should be good.

A standard i3 and a GTX 480 (very good GPU for a LOW price) should do it.

You don't need a good video card or anything like that, but depending if he gets on servers or makes a big world you want a decent amount of ram, 8 gb should be fine.

Minimum Requirements:
CPU: Intel Pentium D or AMD Athlon 64 (K8) 2.6 GHz.
RAM: 2GB.
GPU (Integrated): Intel HD Graphics or AMD (formerly ATI) Radeon HD Graphics with OpenGL 2.1.
GPU (Discrete): Nvidia GeForce 9600 GT or AMD Radeon HD 2400 with OpenGL 3.1.
HDD: At least 200MB for Game Core and Other Files.
Java 6 Release 45.

A decent i3 and a 750 ti should do your nephew some good!