From what I am seeing from most people, the most common classic style recommendations are the Browning 1903/1908, a micro 1911, or a S&W J frame.
I've tried shooting some rental sig 911s and springfield 238s, they felt great but they just loved to jam up on me or fail to go back into battery, the 9mm versions were rather snappy. The were likely just dirty, but it has made a little once bitten twice shy. No real experience with a 1903 or a J frame, no real revolver experience at all really other than renting a 500 magnum with my little brother. Because ya know, why not? I've been hearing a lot of good things about the 1903 on youtube, same with the slightly more fragile but more ergonomic Remington model 51. I won't find those for rent anywhere lol, I'll definitely need to try and rent out a J frame in the future.
As for the newer options, the Ruger lcp series is coming up A LOT. Also a good few sub compact glocks, the XD, and the 365.
Oddly enough, for as much as people love the lcp the one I handled was so narrow it was awkward to hold, though it was a gen 1 not an lcp ii. I shot a glock 43x, while it was fine outside of several failures to fire, I just didn't care for it. Especially the trigger, but apparently that is normal for glocks. I liked the 365 I shot, but similar trigger to the glock and the slide lock was a pain. I haven't tried an XD, but I have shot the XD EZ series and found them to be rather nice, that may be worth a look. A few people also recommended kahr, I've heard great things about them but never seen one in a store.
There are a lot of interesting options to look into that you all have brought up as well, like those little seecamp pistols or the Taurus 709. I'll definitely have plenty of time to research all these as the quarantine restrictions tighten up lol. Truthfully, I really like the r51 I have but after a few range trips filled with stove piping live rounds and the odd failure to extract, even if Remington does fix the thing it would take a lot for me to trust my life to it. It's a real shame, as if was reliable it would be the perfect handgun for me.