Should I be weirded out by my best friend dating a girl who had the same name as his little sister
What is making you feel weirded out?
I've got bad feelings about this. I've got a suspicion that she doesn't deserve him.
It does seem a little weird but judging her that badly is not a good thing. She hasn't done anything wrong and it would hurt him that you believed that.
Nothing comes from nowhere, let me do a background check.
I think you should reconsider that background check. Would your friend be okay with that kind of privacy invasion? The best thing you can do is wish your friend well, and support him if it really doesn't work out.