Phoenix Giants had a longer tradition, but of course the longest traditional name for the franchise was the Seals. The Phoenix Firebirds cap was one of the best-selling minor league caps at one time, so there was definitely some appeal. And in the name, not only is there alliteration, but it's synonymous, like the IL's Buffalo Bisons.
Diamondbacks is okay, but it begs shortening, and "D-backs" is pretty bad. It's not an affectionate shortening like the Jays, Sox, Cards, Pats, Pack, Leafs, Bucs, Niners, etc. etc. As a Suns fan I had no problem with the purple, but the teal was too much of a fad. And yet the switch away from the only major league championship colors the state has...in retrospect I think it was a bad mistake, much as I tried to give it the benefit of the doubt.
I don't remember wanting it to be Firebirds at the time, but also in retrospect I think Firebirds would have been better. Suns, Wranglers, Firebirds, Roadrunners, Rattlers, Coyotes...every single one of those names is better than Diamondbacks to me. (I left the Cards off because being founded in 1898 trumps any name debate for me.)
But I wish every one of them went by Phoenix. I get the Minnesota Twins because they really are the twin cities. Phoenix is the metro area, and that's the way the rest of the country looks at it. It's not a slight to the other cities.