What causes some homies' phones to give a busy signal when called (not able to be a follower)?

As has already been mentioned, there is a cooldown time before you can reuse a homie, but there is also a special case here which isn't commonly known.

You can't call up homies while you're in the middle of a storyline mission. That would cause all sorts of confusion to begin with but the mechanics would also be problematic. While that is perfectly sensible, it causes a problem for the 3 way mission. You complete it once, and you think you're done. You even get the credits up. The game tells you it's complete, but the mission is still there for you to play through the alternative ending.

What it doesn't tell you is that until you DO play through that alternate ending, you're considered as still being in mission. That means that all your homies et al are all busy, and won't unlock until you play through again.

So; if you haven't completed the 3 way mission twice, got the credits twice, then between the first and the second time all your homies will flag as busy.

Hope this helps.

"How long is the cooldown period before you can call a homie again to be a follower after they have been called previously (after being dismissed or after the homie / player character dies)?" "[H]ow long, exactly is the cooldown period[...]?"

I made some measurements in response to When does the helicopter homie respawn?. Unfortunately, I don't have access to the game as I used to, so that's all the measurements I have to show.

Outside of missions and activities, Homies shouldn't be busy. One of my only theories is that you're trying to call them while in mid-air, being hurt, in the process of entering or exiting a vehicle, and/or other such states which may affect call success. I remember trying to make calls, getting the busy signal, and then retrying—once I was truly idling—with success.