This hasn't made the transition from bird to name for me. Wrens are quite attractive small birds but nothing special - and they are very small.
I don't like the way it looks (the first part of the word 'wrench' and Christopher Wren, because of the initial capital letter, are the first things I think of). Then I think of Wrens as in the WRNS - women's navy.
And I don't like the way it sounds - unfinished, as if it's short for Renata or Lorenzo or something.