The process is straightforward. You can begin the process herehttp://www.archives.gov/veterans/military-service-records/correct-service-records.html
To be successful, your son will need to demonstrate, under the circumstances, that an upgrade to the characterization is the appropriate result in the case. Typically this would be done by showing that the result at the time was wrong for some reason...for example if there was some error in the process that was not addressed at the time, that would be a basis for upgrade.
The board will also consider post service conduct, but this to a lesser degree than than your service conduct.
You should start with a request of your records. You can request them herehttp://www.archives.gov/veterans/military-service-records/standard-form-180.html
Unlike the urban legend, there is no "automatic upgrade" for discharges...in fact only a small percentage of requests are granted.
To have the greatest chance for success you son should consider working with an attorney that can help you assemble the package you will submit. A google search under "military lawyer upgrade discharge" will net you several prospects...I understand he can not afford one...perhaps best to save up.
The one thing the lawyer can help with is organizing the information you have to put it into a "package" that will make it easy for the board to see your points. This can be a very valuable service.
BotXXXXX XXXXXne: Its possible but difficult to do. Best of luck
Please let me know if you have more questions and I will do my best to assist, otherwise, please accept so I may get credit for my work.