Hi muminmuddle, Two lovely names! Either way is a good choice. I'll watch this thread to see what you end up going with.
Popularity is not a crime. 'Sarah' is a beautiful beautiful name, for example. So is 'Anna.' Neither is spoilt by their popularity. Likewise, 'Martha' is lovely and strong and feminine no matter how many there are.
I say Jo-zee, btw. Sweet, strong, kind. I've heard of it as a short name for Jocelyn. It reminds me of another name, Essie.
Do the name meanings sway you at all?
I agree with what qumquat: once the name is attached to your little one there won't be a second thought and people will love it because they love her. IMO, I think it's sweet to have existed these few weeks with your new daughter without a name. For goodness' sake, she is not suffering from it! It will make a fun story to tell her when she's older, how you wanted just the right name and how, in the end, you got it. Take your time, get some sleep, and no pressure!
When you do decide, as a reward to yourself and DD, get this personalised storybook: "My Very Own Name". We have it and it is brilliant.