I don't want to go back to my tattoo artist
Did the tattoo artist mess up?
My girlfriend asked me to ink her name on my arm and I don't want that.
If you are uncomfortable of having someone's name on your arm. You should tell your gf that, before it is too late and regret it later.
I should ignore it?
No, its your body and you get to decide what to do with it, but let the tattoo artist and your girlfriend know.