Sometimes, but mostly just to give them nicknames and then they just keep going by the nickname. For instance my current challenge Sim's wife is named Alessandra (French) Conrad, but she goes by Sandy. I got her from the Gallery. Their son, Cale Jr., used to go by Junior, but now that he's older, he prefers to be called CJ.
My challenge Sim from last year started out named Tessa Tyler. But that wasn't her real name, she was in a witness protection program. Over the months her name changed from Tessa Tyler to Vanessa Varela and finally her real name, Essa Ellington. Well, she's Essa Hunter now since she married. Essa got that name from a suggestion by one of the Simmers doing the challenge. She said it meant 'star' which fit perfectly for my Sim and her story.
My Drifter founder started out with the name Elle McDonough, but I accidentally failed the first time I tried it by letting her marry one of my Super Sims. Not really cheating, but I didn't feel right about it. So I started over using the same founder, but called her Ella instead.
Then there's the case of my Hunter Brothers; Joel, my first Super Sim and the oldest, Josh, his brother I made from cloning Joel and then changing Josh a little bit, he's a Super Sim too I used for the Asylum Challenge and in my own challenge last year, and Jon, the youngest brother. He started out as a Sim from the Gallery named Alex Hastings. I modified him for a challenge here on the forum and that's how he was renamed and ultimately became the baby brother of my Super Sims.
Lastly, I've changed a bunch of game generated Sims' names because I couldn't pronounce their given names. But I only bother to do that if they mean something to the Sim(s) I'm currently playing in a save. Or, if they were a Sim generated for a role, such as a Mail Carrier, Gym Trainer, etc. and I liked them so much I wanted to keep them for one of my custom template saves to do those roles. Usually they need a makeover which sometimes includes a new name.
I think those are the only incidences where I've changed the names of my Sims, but I've had so many over the years, I suppose there could've been more. :D Great topic OP!