Tanya Murray of Canada was excited this summer about her family’s pending trip to New York City. Except for the rates at the Midtown hotel she had booked: $1,600 for six nights. “That’s crazy,” she says. “I was having a heart attack about how much the hotel was going to cost.”

Happily, she found a much cheaper alternative: Camp Hudson in Staten Island, a National Parks Service site. Last week, after driving 32 hours from their home in Regina, Saskatchewan, the Murray family pulled up in their red mini van and pitched a tent...