I'M SORRY BUT YOUR MOM IS FUCKING RIDICULOUS.

I would not allow them inside the house or on your property and have police waiting on scene. There’s a line and they have crossed it. I’m sorry, OP. Consider getting a restraining order.

That would make him your daughter’s first cousin, once removed. He needs to be removed again — this time from your house right now! But, seriously, folks, let’s focus on what’s really important here: Did he being her a really nice gift?

I'd consider a family upgrade, you know, preferably without the crazy factor.

To use a dated cliche, "Awww, hell no!" Kid's birthdays are never a good time to handle ANY family drama, particularly a kid that young. Also, when someone holds a member of your family hostage, no one has to apologize to them and you definitely don't invite them to the birthday party of the family member they held hostage!

By the sound of this story, I take it you never filed a restraining order against your cousin after that stunt. If you didn't, now would be a good time to do so. And I recommend cutting your mother out of your life.

No! No! No! No! No! Dump your idiotic, victim blaming, mind game playing mother and tell evil cousin the cops will be called if you see him anywhere near you or your kid. Failing to return fifty bucks you loaned him- sure, maybe he could expect a second chance. Puts your kid in danger- no second chance. That is a one-strike-&-you're-out type of thing if ever I heard one.

is there any history of drug use in your family

I would draw a line in the sand and tell her that either she retract his invitation or other arrangements for her birthday will be made. She is your daughter, not your mothers and as such you get to make the decisions.

Oh, hell no! You need to make it clear to your mom and to your cousin that he's not welcome at your house at all and that you are not about to apologize for having a man imprisoned after he threatened your daughter. Hopefully it's not too late to get a restraining order against him.