I recently found on a website a picture from a gathering. An individual had a sign saying She's someone's sister. As a non-native English speaker, I'm wondering why isn't She's someones' sister the correct version?
Someone's sister is correct. 'Someone' is singular, as it refers to a singular (unknown) person. "She's Sarah's sister" and "she's someone's sister" are similar things. e.g. you don't know who it is so "someone" has a sister.
You didn't give much info but I think the sign "she's someone's sister" means something like -- she's the sister of 'someone' which could be you, your brother, your parent or whatever. In that case it would mean "she could be the sister of someone close to you [implied: and how would you feel about that]".