Don't worry if it doesn't go with Claire as middle names are hardly ever used after the first couple of weeks of introducing her to everyone, the only people that will ever see it again are you, her and passport control. Most forms you don't even have to include it.
Ranked in order for my preference:
1 Polly - Absolutely lovely, my 1st choice of your names!
2 May Absolutely love this name, it is a family name for me and I hate that it ahs become a filler middle name as it is really special to me. It was in my top 5. It only comes second to Polly because of all the annoying people that boringly used it as a middle name!
3 Nancy - Cute.
4 Martha - This was fairly popular round my way about 10-15 years ago, I find it a bit dull and know a handful of little girls called Martha now.
5 Connie - Cute, bit 50 year old woman, and I'd assume parents were country music fans.
7 Hazel - Seems to be pretty popular on here, I find it really unpleasant sounding, reminds me of a Mrs Trunchbull type character. Sorry!
6 Lydia - I'm not sure if it was your post or someone else's but I mentioned that bizarrely, this name makes my skin crawl, couldn't remember why at the time but I do now - Aunt Lydia from The Handmaid's Tale! She's a truly terrifying woman who commits some unbearably cruel things on the women in that story, but a lot of people have not read the book or seen the TV show so you'll probably be ok!
Good luck!