We found most methods that intend to be fair, are.
In the rulebook it says to use whatever method you like, but the suggested default is that after roles are handed out, everyone draws two characters at random and chooses one. Gettings roles first is important beacuse a handful of characters (read:Eirin) are distinctly better at some roles than others.
When teaching the game we aways use the 7 simple characters. Those being Reimu, Marisa, Tenshi, Patchouli, Utsuho, Remilia, and Youmu. (Edit: Youmu, not Satori, derp)
A common houserule is that if your character survives a game you can keep it for the next game.