Wow, I'm loving all the opinions about this. Okay, I'm going to try and respond to everyone in one post.
Proud2BeAnAmerica - DD has a peanut allergy too. It's funny you mentioned that, I hadn't even thought of it. I don't like the pronunciation 'Reece' for us anyway, though.
wookie -Thanks! That is exactly how I feel about the spelling 'Rhys'. It just has such a great feel to me.
RedOakMomma - No apology necessary. The only 'joke' I can come up with is Rice-o-roni....which actually isn't that bad IMO. Every name has potential to be a cruel joke. At least this way, kids wont' have to be too creative in coming up with something really dirty or offensive.
JessieBird 
Looks like we have the same taste in names.
On names that have been suggested:
I like Everett, hadn't thought of that one yet. Thanks for the suggestion.
Ryder was on my list when DD was still a DC of unknown gender. However, I can't use it because my ex and I picked it out and that would just be weird.
Wesley is a nice name too. It reminds me of the Princess Bride.

I'm just not sure if it's
this little one's name.
beanbean I would have never thought of the association of 'white rice' as a negative thing. But maybe others will. The food names you mentioned didn't feel all that different to me. It makes me think of the celebrity who name their daughter Apple. When I first heard that I only pictures the fruit in my mind and thought it was silly as a name. But I think it becomes a name if that makes sense. I think Rhyce sounds much less foody than Apple fwiw, and she certainly got away with it. I agree about the sound outweighing the spelling for sure. Thanks for the tips!