Leah had a mole like that removed when she was 9 or 10 -- it was on her cheek, far back close to her hairline. The dermatologist sent her to a plastic surgeon for the procedure. She was fine -- the mole just looked a little iffy so taking it off was better safe than sorry. She has a tiny scar & had to skip swim team for 2 weeks or so. I don't think it ever occurred to her to ask for it back!
I had my tonsils out in kindergarten and never thought to ask for them. My youngest sister had hers out in 8th grade and brought them home in a jar! I was a little peeved since no one had offered to let me take mine home all those years earlier.