We went first weekend in May, specifically to target the best weather. Fall would be ok, too, possibly late September.
Temperatures are similar to Grand Canyon -- highs of 100+ in the summer! Check the Average Temperatures
for Phantom Ranch, as elevations are similar.
The village is named Supai, the general area is named Havasupai. The stream is Havasu Creek. The two main waterfalls are Havasu Falls and Mooney Falls. Good luck keeping all that straight!