I handle criminal appeals on a regular basis and would happily discuss this issue with your friend. He can file a writ that allows him to use evidence outside the normal record on appeal. Most of what you wrote was probably not considered in his appeal because on appeal, you cannot go outside the record. Please contact me for a free consultation.
If you show up to court ready to cure the mistake and take care of it, they will likely treat you fairly and will set aside the warrant. They might not, and instead might arrest you at court. So it is good to have an attorney make the appearance for you, and get everything straightened out. Many attorneys are at the court house already, and will work with you to make it affordable.
You could file in small claims, but if they show up, it's your word vs. theirs. If they don't show up, you'll have a small judgment and will be tasked with collection that might end up costing more than the fine.
You can file an appeal, but that requires a substantial bond. If you had an attorney, you should discuss your options with him. If not, you should contact an attorney immediately as there are deadlines to file an appeal.
It is possible to unwind a guilty plea, but the motion is complex. You should contact an attorney to weigh the cost/benefit of doing so. Your son may also be able to apply for a restricted license that will allow him to travel for work or school.