Beck/Becks to me is a nn for Rebecca.
I suppose people would think you were fans of the singer perhaps?
Sounds very nick name like to me, I'd not like it as a full name personally, and Beckett to me is not a name, the only association I'd have is with Samuel Beckett the poet/playwright or perhaps Thomas Becket the murdered Archbishop.
But then I know that people like using surnames as first names (I know some children with what I think are very odd names like Tanner and Cooper) so perhaps it would be just accepted as mainstream now.