When I read this part of your message (“She reports that he fixates on words and phrases, for example he may hear a joke in conversation and then repeat it again and again”), it occurred to me that X may be using self-talk to regulate himself or to process the joke that he finds hard to understand.
However, the next example you gave did not fit in with this! I feel there is a lot missing from this picture. I would like to know about his emotional regulation and relationship dynamics when he is engaged in this repetitive speech. Unless we know more about the context, we could probably all come up with a myriad of possibilities.
In my opinion, for a teenager with a PDA profile, he is going to have some attachment dynamics and emotional defences that will need careful consideration, before attempting to teach him any modifications to his social communication.
Some questions I would be exploring with his mum would be, how is he behaving and relating at these times, eg does he seem engaged with her and seeking connection or in his own world, is he relatively calm and regulated or is he in a state of resistance? Invite his mum to talk more about his patterns of communication with family members and wherever there is a familiar working attachment (although I am sure you have already done this).
Other ideas, is he experimenting with language in a playful way? This would be interesting to explore in a session by making up sentences from words given, (like in the sentence formulation subtest of the CELF).
It’s too easy to go off at a tangent when there is not enough context! Forgive me for doing so, but hopefully there are some ideas for you to consider.