Giving birth after 35 (in this case 37) is statistically more risky than offspring from first cousin's. A third cousin is much more distantly related than a first cousin, your first cousin has a pair of the same grandparents as you, your third cousin shares a pair of your Great-great-grandparents.
It could be a recessive trait (1 in 65536 doesn't mean "impossible")... but it could also be one of a near-infinite number of other possibilities.