1d4 Is Not Enough

My daughter's friend was rolling up her very first D&D guy today! Yay! She has a new Chaotic magic user who has a goat named Death and a mean streak.

She rolled hit points and got a 1.

"Nah, do it again."

And a 1.

"Again..."

And again and again until she hit the jackpot: 4 hit points!!

Um, that's still really lame!  In my case, I would have rolled with the 1 hit point, cause I don't mind death (the existential state or the goat). But for a noob, being a level 1 wizard with one spell seems to be awfully punishing.

YES I know the proper methods to avoid death (theoretically at least) and so forth. But it still seems unusually punishing.

I don't have a solution to the problem that I like.