I don't know the meaning, but I thought I'd give some usage statistics since it might help everyone else.
http://www.meertens.knaw.nl/nvb/naam/is/Gaite
It appears to be exceptionally rare.
http://www.babybytes.nl/namen/meisjes/Gaite
I think this site states it was first recorded in 2011? (which is possible even with the actress; since unique names will often go unrecorded if they are not given to enough people for privacy reasons).
No idea what it means, maybe a feminisation of Gaius?