Some good thoughts. In the absence of an individualized good reason not to, I would let the patient record. In my mind, it is their session to do with what they wish. If I felt someone's competence to weigh the risks and benefits of protecting the information was impaired (for instance, they are manic and want to post it to a YouTube channel to prove to others that they are Jesus) then that would be a different story.
There are some real pros/cons, though.
Cons:
-They could post it to YouTube, or anywhere else. They could post clips out of context, or even doctor the clips in an unfavorable way.
-In case of an adverse event, those recordings would be admissible in court and might be used selectively to make you look bad.
-There could be psychotherapeutic issues as mentioned above (for instance, the patient "taking you with them" in a somewhat regressive way).
-It could make you anxious or self-conscious, and could thereby impair rapport.
-The patient could accidentally leak it, and even though it's not your fault that might be harmful to them.
Pros:
-It could serve as an aid to memory, because the patient could review content that might have flown over their head the first time. This could be especially important for patients with impairments in memory, attention, verbal abilities, etc.
-It could improve rapport; you are granting them a potentially uncomfortable request.
-Recording might make them feel more comfortable, and more at ease that you have nothing to hide (for instance with a paranoid patient).
-It respects patient autonomy, emphasizing that it is their session and respecting their process (presuming that the patient believes they can get more out of the interaction by reviewing it later)
-In cases of any alleged misconduct, such as sexual misconduct, I would personally prefer if a recording existed that a third party could review. In other words, in some cases a recording could be protective.
You have to know your local laws, and weigh out what feels right for the patient in front of you. I agree with the above posters that surreptitious recording is not okay. The treatment relationship is based on mutual trust, and that would be a real violation of that trust.