from Wiki:

"The official commentary included in the CD set Celebrate: The Three Dog Night Story, 1965–1975 states that vocalist Danny Hutton's then-girlfriend June Fairchild suggested the name after reading a magazine article about indigenous Australians, in which it was explained that on cold nights they would customarily sleep in a hole in the ground whilst embracing a dingo, a native species of wild dog. On colder nights they would sleep with two dogs and if the night were freezing, it was a "three dog night".

I recall hearing this 40+ years ago.