1999 Chevrolet Cavalier A/C problems

My daughters a/c in her car doesn't work when you push the button on the control panel but a mechanic jumped the wiring harness and said the a/c works. He said its something wrong in the computer not telling the a/c compressor to work. I have asked other mechanics and they said it could be the computer the wiring or the control. She is in college full time and works part time so she doesn't have much money to spend at the shop. I am trying to help all I can. Would just like to know what direction we need to go without spending a ton of money.