My son is currently playing Minecraft on an i3 2120 with on-board graphics, no problems.
Some of the settings are turned down, but it's perfectly playable.
If it's just for Minecraft I'd buy a cheap dell, hp or similar off ebay and stick a cheap graphics card in there.
The system requirements are:
Minimum Requirements:
CPU: Intel Core i3-3210 3.2 GHz / AMD A8-7600 APU 3.1 GHz or equivalent
RAM: 2GB
GPU (Integrated): Intel HD Graphics 4000 (Ivy Bridge) or AMD Radeon R5 series (Kaveri line) with OpenGL 4.4*
GPU (Discrete): Nvidia GeForce 400 Series or AMD Radeon HD 7000 series with OpenGL 4.4
HDD: At least 1GB for game core, maps and other files
OS:
- Windows: Windows 7 and up
- macOS: OS X 10.9 Maverick
- Linux: Any modern distributions from 2014 onwards
Internet connectivity is required for downloading Minecraft files, afterwards offline play is possible.
*Current minimum OpenGL requirements is 1.3, but modern GPUs often ship with newer versions of OpenGL
Recommended Requirements:
CPU: Intel Core i5-4690 3.5GHz / AMD A10-7800 APU 3.5 GHz or equivalent
RAM: 4GB
GPU: GeForce 700 Series or AMD Radeon Rx 200 Series (excluding integrated chipsets) with OpenGL 4.5
HDD: 4GB (SSD is recommended)
OS (recommended 64-bit):
- Windows: Windows 10
- macOS: macOS 10.12 Sierra
- Linux: Any modern distributions from 2014 onwards