As the tittle says it. Can I say "My mom's kids names" to mean: "The names of my mother's kids" ?
Not quite. It should be "my mom's kids' names." You need two apostrophes because there are two possessives here: The long form would be "the names of the kids of my mom."
Some people feel that chaining possessives in this manner is informal and would recommend rewriting it as "the names of my mom's kids." However, at least for informal language, people can (and do) chain possessives.