A captured orc joins the party and rather than become a traitor, he sticks around and becomes a henchman.
A player wants to play a role of humanoid wolfmen he came up with.
Someone is reincarnated into a kobold and wants to continue to play the character.
While I have an idea to present a list of "racial traits" to help make these things easier (and to help pave the way for random race generation!), the enterprising DM need not fear or reject these situations.
Think about the situation and simply assign 3 or so abilities to the race. Elves are stealthy, can cast spells in magic armour and see in the dark, for example.
Pick only things that are significant, interesting or very obvious. The wolfman probably has a keen sense of smell and a warm fur coat, while kobolds can squeeze into tight spots and are immune to digested toxins in your world.
Classes are probably self explanatory but don't be too limiting. If all else fails, most everyone can be fighting men or thieves. Set level limits where you feel they make sense. If you believe in the "old school" nature of level limits, custom races should probably be slightly low, but that's for the individual DM to decide.