One option is to make them small and insignificant enough that they don't really contribute to battle, but provide some other subtle benefit.
When playing with younger kids they can really get into the idea of having a familiar but they will lose their $#!t if their familiar is hurt or dies... so you best have an alternative
I've got it setup that when we use familiars they will never be targeted by Monsters unless there are no other Heroes in the same room or corridor. What's more, we usually have them be small enough that they can choose to "hide" with the Hero that owns them. When they are hidden they are removed from the board and can't even take damage from AOE spells, etc like Firestorm, etc. and we say they're protected within a backpack or under a heavy cloak or what ever. If they would get targeted in combat they're smaller than Heroes usually so we make Monsters roll a Red Die and say they miss completely on a 1-4. If they roll 5-6 then the attack happens like normal.
The Stoat we've got at the moment is setup like this and can squeeze under doorways, revealing what's in the room, at which point it can squirm back out again. Usually we're close to: 8 Mv, 1 die Attack, 2 dice Defend, 1-3 BP, 1 MP. Pixies/faeries are good choices for Elves, and we've used those.
We've avoided anything larger like dogs because I tell the kids if they take a dog in it's going to be big enough to get into fights with Monsters on its own but will probably die in battle with a Goblin. That's enough for them to leave it outside the dungeon
~N