I've liked
Lydia for a long time for its gothic sensibilities, although it's always bothered me that it doesn't have a satisfying meaning!
I love the idea of passing your middle name on, so
Lydia Margaret is my favourite of your two options, but I think a (typical) child would probably enjoy being called
Lydia Jade more. I'm not a fan of
Jade, though, and would prefer
Lydia Jane, but that's just me.