I don't agree there Jimmy, calling her naive for wanting 'root'. She isn't thinking of a plant root, nor are you and nor am I. the OP will have to look for a Japanese translation, or she could use the term 'Japanese blood', which, if I am right, would be three symbols, which shouldn't look that bad.
Also, if she is definitely wanting it, I doubt that it is a long phrase would deter her. The action of wanting to express her roots in tattoo format doesn't necessarily mean she wants a beautiful tattoo, just that she wants one.
I say to the OP 'go for it'.