I would start by using the
Recommended System Requirements from Mojang and price out a computer that gets close or exceeds that from there. If you're looking for something that will last longer and play newer games you'll need something much better than that though.
Quote:
Recommended Requirements:
CPU: Intel Core i3 or AMD Athlon II (K10) 2.8 GHz
RAM: 4GB
GPU: GeForce 2xx Series or AMD Radeon HD 5xxx Series (Excluding Integrated Chipsets) with OpenGL 3.3
HDD: 1GB
Latest release of Java 7 from java.com
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I assume you can get a computer that meets those specs for around $500? Double the cost if you want something that will last a while.