Counterfeiting can be charged as a federal crime (18 U.S.C. §472) with possible imprisonment as high as 20 years or as a state crime (Texas Penal Code §32.21(e)(1)) with possible imprisonment as high as 10 years. Either way, the prosecution must prove that he was in actual possession of the $20 and that he had knowledge that it was counterfeit. Possession is a big picture argument, and depends on several factors including where he was/where the $20 was, who else was within reach of it, etc. While we get in trouble if we guarantee the outcome of a criminal case, I will tell you this sounds like it is a case worth fighting, and I would not recommend he plead it out based on the information you provided. If you choose to call my office, I would like to talk to you at no charge and find out more information so I can more fully answer your questions and let you know what representation would cost if you or your son choose to have me represent him.
As noted above, it is impossible (and inadviseable) to give a guaranteed result, however there appears to be more to the case than a simple quick plea warrants. Have the case investigated, and if you can, hire an attorney if you feel the Court Appointed Attorney is not personally invested in the case. Court Appt'd Attys are very competent practioners but you are free to retain an attorney that you can involve in the investigation.
Possession is a loose term, but from the facts above, your son might have a solid defense.
Although my intent in answering this question is to aid you in the legal process, my answer does not establish an attorney-client relationship in any way. You should seek the advice and counsel of a qualified attorney in your community to evaluate your legal needs and to advise you. No Attorney-Client Relationship is created without the specific intent of both parties.
Possession can be joint or by one person. Generally if no one is holding the item, it is "assigned" to the nearest person (or whomever the cops claim made "furtive" gestures towards the area where the items was found.) If there is only one person in the car, then that person generally gets the charge.
Some of the "affirmative links" that are considered are: 1) driver of the vehicle; 2) owner of the vehicle; 3) who has the keys; 4) who is nearest the item; 5) is the item hidden or in plain view; 6) can the item be smelled; etc. There is no exact formula.
It is possible to get fingerprints off of the bill. However, if your son ever touched it, then that would just seal his fate. (It does not matter when he possessed it - just that he did.)
The lawyer probably fears the "defense" in the case - "my dealer dropped the money in the car". It would be a bit difficult to sell but it could be done with the proper voir dire.
If you are not satisfied with the appointed lawyer's representation, then I suggest you invest time and money into finding a lawyer with whom your son can be satisfied. The lawyer may have the same recommendation but at least you will be more comfortable with the decision.
Although I have answered the question to try to help you, you should consult with a lawyer in your area in person on the matter. In addition, my answer does not establish an attorney-client relationship between us.