Rock climbers and boulderers refer to those rocky hills/cliffs/mountains as "the Gunks." Famous challenges for climbers, but, since we are on the topic of family places with the de Medicis and the von Trapp family, we should not omit the Mohonk Mountain House. I have a good photo somewhere of my Grandpa fly-fishing on the lake.
It was a favorite of my grandfather, and remains a family favorite. Nowadays, they even serve alcohol which the Quaker Smiley family never used to permit (you had to sneak in your own, and secretly imbibe in the privacy of your room before dinner).