Change your name... where? I don't think you mean here, unless your sister-in-law's last name is "lemmy.world". If you're talking about a platform (like Facebook, Google, etc) you might be able to do it through them. But it would depend on the platform, and would be different for every single one...

On the fediverse, from what I read, no, name is final. Apparently that's because in the fediverse, there's no underlying user ID, so an instance checks the user at (@) a given instance, not someone else for the same contents. And forcefully replacing could cause issues, going by some instance I'd seen that changed the software soon but not enough, making connected instances have issues handling content mismatch.

Also a bit of a tangent, but if it's also your last name, I don't see what's the problem of using it? Multiple people may have the same one, so there isn't an exclusivity on name usage.

And if people are picking on her because of the shared name, see if people doing so reveal their names. And if not, ask them to, to either disarm them or reveal their intentions.

Nope, you need to make a new account.

A new one? that sucks but understand much love.

Yes. Otherwise it breaks federation and any and all of your posts and comments will only be available on your local instance, and viewable only by local .world users. That is if the admin is even willing to make the change.