Making a Music Video

I've never shot a music video, but performed in quite a few. (In my youth I was a touring musician.)

Generally, if you are doing a "performance" video, you get an audio playback system (their PA would do nicely) and blast the studio recording. You record the on-set audio as a sync reference.

You'll get a better looking sync if they really play and sing hard, as if they were doing an actual performance.

One of the few credits to my name was actually a performance music video.

We played the music over a stereo and the guitarist/bassist played with their amps off. Make sure you record the live on spot audio too - for syncing with the studio track.
For the drums I had the drummer put an earphone in and wire it down his back. Making sure to either keep the ear facing camera clear, or keeping the camera far enough away so that you couldn't tell. I'm sure playing the song over speakers would have worked but we couldn't get it loud enough so that he could play and make it look realistic.
With the wide shots the drummer had the earphone in and the band played along to him.

And I suppose this applies to any shoot but - biggest advise: roll with the punches. We somehow forgot microphones on the day of the shoot - so we stuck a flashlight in a mic-stand and the singer sang into it. Even got close-ups of it and one (rather cool) shot of it from the singers point of view.

If the boom mic is going to be in shot - you might as well go full monty-python and the holly grail.

Fantastic, thanks for all the suggestions. I'm no stranger to recording music itself, a few of my songs are on iTunes, so I think I might actually make a test music video to one of my songs. I think the band I'm doing it for might actually want a story type music video, where they act in it (something like "You're Beautiful" by James Blunt music video) in this case, I don't even need to really worry about the audio, I'll just film it like a regular short film and sinc it up with the audio later.