Personally, I do think it is dated. It brings up images of the late 70's and the 80's. I don't like Bethany either. If it has significance to you though, I don't think it really matters. I prefer Elizabeth, Elspeth, Elise, Elsie, Elsa, etc. better.
I agree.. anything but Beth. I just dislike the name, I think it's very bland, along with Bethany.On the other hand, it IS for your aunt and mother, so I don't see why not. If you like the name, go for it! It doesn't matter what anyone else thinks.