I cracked the hinge on my last cheap phone, so I bought a new cheap phone. Even though this one had one decent ringtone that wasn't the same one my wife uses, I decided it was worth it to have a song as a ringtone despite having to pay more for a few seconds of song than the whole song would cost.
My criteria for a good ringtone is primarily about hearing it when it rings and distinguishing it from other phones (especially my wife's) and only secondarily about being a cool song. There should be a lot of contrast between the ring and normal ambient noise., and within the music itself--In the same way that Beep----beep----beep is easier to hear than beeeeeeeeeeep.
On my last phone, I had the intro to Baba O'Riley. It worked well, lots of staccato tones, but discovered that the beginning of a song you hear relatively often isn't the best ringtone either--I reach for my phone when that song comes up on my playlist.
I had a lot of songs in mind that had parts that I thought would make good ringtones, and many of these songs said they were available for my phone--however the vast majority of them used the vocal chorus, completely ignoring a fantastic and easily-recognizable instrumental riff.
I wound up with the horn section of Low Rider by War-An OK but not exceptional song, but works very well as a ringtone.