I don't feel like hanging out with my friend on her birthday tonight.
Why won't you hang out with your friend on her birthday?
I have other plans for tonight
It's okay to celebrate a birthday on another day. You should text or call your friend and wish them a happy birthday. This will show them that you appreciate them.
I'm going to give her a gift and I'll text her.
I am sure she will appreciate the gift and the warm wishes. What other even were you planning on going to tonight?