The background Urchin provides the player with "a pet mouse". Is there any reasonable way to make this mouse survive combat? I would think that the first area of effect spell (say fireball) would kill this poor mouse instantly, even if it was stuffed away in a pocket. I see no point of giving the character a mouse if it either must be set free at the beginning of the campaign or die in five minutes in a flaming ball of fire? So dear internet, is there any reasonable way to keep the mouse alive?
This is one of those classic "already-been-solved" problems:
Mice in the wild face existential threats every day. They've already figured out their best strategy: find very safe places to hide, and scurry among them when necessary.
I don't see why this would be any different in a D&D-verse. The urchin's mouse--during fabled background times--will have figured out the good places to ride along. Wedged between a waterskin and spare jerkin in the pack? Between layers of a tied bedroll? We may find those locations claustrophobic, but that's comfort to a mouse.
Even if combat kills your poor urchin, I'm sure the mouse could be fine. Unless caught out in the open. And that's the life of a mouse.
There was much made in comments of whether the mouse could possibly survive the hypothetical fireball in OP. Please see Would a very small creature inside a sealed container carried by a character take damage from AOE spells? for the correct explanation of how it could.
As a frequent DM of 5e AL games I would actually rule No, fire ball would not hurt the mouse. While strictly speaking the pet mouse would be a creature, it is there purely for flavor and can not actually help in combat or the like. In this manner I would treat the mouse like a trinket. So I would not kill it off unless the player wanted it to die to give him/her more of a background.
Yes if you let him in your pocket. Besides that, a pet mouse can be really helpful... For example rats are particularly gifted at running through mazes. Their maze-running ability comes from their evolutionary history: rats are small burrowing rodents that have spent millenia digging and finding their way around underground tunnels. It's no wonder they have a knack with mazes.