The noun being referred to is "nicknames" (needed by "them"), which is plural.
So in this context, one nickname is needed for each of the people in the "them" here, and if she is making one up for each member, then it would be "So she made them up." But I would find it more natural in this case to write it as "They needed their own nicknames to join the party, so she made some up.".
If however, what's being said is that she- as a member of these people- is only making a nickname up for herself, and the others are doing so by some other means, then it would be "She made one up (for herself)."
Otherwise, making "one" up doesn't make sense here, since the noun is "nicknames" (plural), and "one" is singular (same for "it"). But if they needed one nickname for them as a group, then it would be "so she made one up".
"She made ones up" might work, but it just feels wrong here (as a native speaker). I can't put my finger on why. It's not that "ones" isn't a really word or can't be used in slightly similar phrasings. For example, one phrase using "ones" I just randomly though up is "Yes, officer. Those are the ones right there."