These are most interesting situations, aren't they?
I am curious ... why would you need this to hold up in court? Are you thinking you need to evict this Resident? It should not come to that, I don't think.
What you can do is:
a. Talk to the mail carrier, but lately there is less consistency in carrier assignments and often there is not the same one. Besides mail carriers are not supposed to provide you with this information. (Although I have been known to follow the carrier to the box and peek at names as he/she places it in the box because they cannot "tell" you the information.
b. Mail something yourself to the unit in the name of the son. If it comes back as undeliverable, then there is your legal proof, probably a better option.
c. Declare an emergency in the unit (water leaking next door, need to check the unit) and go in to see if there is clear, visible evidence of the son's presence: clothing out, his shoes lined up by the door, toothbrush on the sink, shaving supplies. If there is, you send a letter to the mother letting her know you suspect her son is living there and that she needs to bring him to the Office to fill out an application immediately (but give a reasonable time frame of two days/48 hours if he is 18+ years old.
d. Once you have his Application, process it and see whether or not he is qualified.
e. Call the police and have them let you know if the car is legally registered and plated and inform the car's owner to move it NOW, within 24-hours or it will be towed. All Resident cars should be registered in your office. Have it towed after that period of no one complies with your request.
f. Let your Maintenance team, courtsey officers and polic department know you have suspicions about this person. However, I trespass people off properties all the time and have no qualms whatsoever about it. I would consider this if the young man personally told you he has a violent criminal past.
Regarding the Resident's transfer request: she has already signed her lease????