For me, Sons of Apollo is a bad name on all levels. First of all, 50-something men calling themselves "sons" rubs me the wrong way. While there's nothing logically wrong with it, it still spawns a dissonant image in my brain. Also, if ad absurdum the band made it huge, they'd probably be referred to as just "The Sons." Secondly, Apollo is a way too generic figure for it to have any significant meaning. I mean, I get the thought process, he's the god of music (and a million other things), whoopty doo. Still meaningless.
Compare it to Children of Bodom, for instance, a similar name. They were basically kids when they started out, so "children" is fitting. "Bodom" is way more specific and personal, it ties in with their home country, has a sinister backstory to it which the band captured with their sound and lyrics.
When it comes to good band names, the most important thing is that is has to have a zing, it has to sound cool. The meaning of the words is secondary, it can be gibberish, but if it sounds cool, it's a good name (Metallica, Megadeth, Led Zeppelin). Try saying those out loud, they have a nice ring, rhythm and flow to them. Saying "Sons of Apollo" out loud makes me cringe.