She is a cousin of Claire's. or She is a cousin of Claire.?

Answer 1

Either form may be used.

The first form is more usual, that is ... of + possessive.

e.g. 'He's a cousin of the President's.'

There is a variation where the noun does not have the possessive 's, which is sometimes used when talking about relationships, particularly where slightly more emphasis is placed on the last person mentioned. e.g. 'He's a cousin of the President.'

[See Practical English Usage, by M Swan, Oxford University Press, section 434. "Possessives with 'of' "]