One of my laptops stopped turning on. Based on the grinding and buzzing sounds it had been making, and the elevated temperature readings at times, I'm pretty positive the fan failed. If I decide to take it somewhere seeking to have the fan replaced (depending on how much I find that it costs), does anyone local in the Phoenix, Arizona area have a recommendation for where? (I might try to take on the challenge of replacing the fan myself, but I don't think I have the dexterity to do it without damaging the laptop.)

My default would be Data Doctors, partly because I trust them but partly because I don't really have experience with anyone else for computer repair.

While they're horrendously overpriced, I'd probably say Data Doctors is one of the best solutions I know of. Sadly (in this circumstances), I always did my own repairs personally so have never tried any of the local places, and at work have always had onsite repair from the manufacturer.

Well, that does tell me Data Doctors would very possibly charge more than the hundred dollars or so that I probably paid for this used laptop.

Open it up, clean it.
Grease the fan (just the bit in the middle obviously).
If you're lucky it'll start working again.
But even otherwise, those are easy to replace and cheap to buy.

Very good chance of it. I think they charge $120/hour for hardware repairs, and minimum is 1/2 hour charge.