My only concern is - will she like it? Sparrow is a name she would either love or hate - there's no middle ground. It's a big gamble. If think the pros outweigh the cons, then I'd name her Sparrow Elizabeth. If you have concern she may not like being named Sparrow - I'm thinking of how much she'll want to fit in with her peers as a child and a teenager, and I'm thinking about her walking into job interviews - then I'd name her Elizabeth Sparrow, and call her Sparrow as a nickname.
I strongly suggest naming her Elizabeth Sparrow and calling her Sparrow as a nickname.
I hear the sentiment, and your difficulty conceiving certainly exacerbates the need for a name that is very meaningful and especially loved by you.
Now the name itself, Sparrow, isn't my taste. It's not a lovely bird - to me it's dark and drab, like Blade suggested. If you really love it, I'd use it as a middle. You can call her Sparrow if you want, but she'll have something more classic and mature to fall back on if she prefers it.
If you're set on Sparrow, how about...
Someone else mentioned Sparrow Adelaide which i think flows really well, and like heaps of people have said, she can always go by her middle name if she deosnt like it.
Like Sparrow as a first name, not one I would use but I'd love to see it on somone else's baby :) Howeber I don't like Elizabeth as a middle name, it's boring and "filler" in the middle, but I love it as a first name. I like Elizabeth Sparrow or Sparrow Eliza much more :)
Other suggustions (sorry for any repeats)