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I'm currently working on cover letters to a public defender's office for an internship - I was wondering if it is considered inappropriate/juvenile/out of place to mention that my interest in defense work/criminal law first stemmed from growing up with a mother who does PD work.

It's where I'd want to jump off in the cover letter to explain my interest in their office, but I don't know if mentioning a familial relationship that's not my own accomplishment would come off as unprofessional. Input would be appreciated!

That's not even close to tmi

It's fine to mention that. It's one of the reasons you have an interest in this type of work, and it's probably not obvious from your resume, so putting it in your cover letter makes sense.