Outside of the practical question, I have a nostalgic fondness for the original JLA Satellite and the classic upside-down rocket ship Super-Hero Clubhouse of the Legion. The Moon-based Watchtower of Grant Morrison's JLA was pretty cool as well. And I think the futuristic Legion HQ of the classic Levitz/Giffen years--with its antigravity architecture, storage tesseracts, and tamed Computo to run the show was pretty awesome, too.
Yeah, the Levitz/Giffen Legion
run is one of those times a group's HQ has really come alive for me. The story where it gets invaded by Computo (annual #1, "Monster in a Little Girl's Mind") is one of the best comics I've ever read.
I thought the giant headquarters the Canadian government built for Alpha Flight during Fabian Nicieza's run was really cool. Shame that when Scott Lobdell took over, he leaned heavily on the hoary and boring "the public hates superheroes" trope.