Me and my wife have been together for 13 years and we both got our first tattoo together. I would never get her name tattooed on my body because the way I see it (and I've seen it with a lot of couples) tend to breakup after getting them. May not have any correlation between the two, but I feel that getting each other's names tattooed is a kiss of death. (again, may not be true but it's an assumption I've made after seeing it happen to other).

There could be a lot of reasons she does not want to get a tattoo with your name on it. If I was her and you kept on pushing the issue I wouldn't do it just out of principle. But maybe she sees it as you wanting to brand her and claim her as your own. That's how I would see it. It's like having a permanent engagement ring and a way to tell someone that you are taken by someone else.

But again, there could be lots of reasons as to why she doesn't want to get your name tattooed on her body. To be fair, it is her body, not yours and she is not your property to force her to do so, and if she doesn't want to do it then she doesn't have to. If this is important to you I suggest you sit her down and talk to her about it.