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I'm a relatively green DM. I've ran a half a dozen games with some decent players. A couple of other friends heard and asked if I could be a steady DM for them. One of the players brought her little sister who has never played before. We spent one night going over character building and the like. Everybody went home with a character they liked and started working on backstories.

As a player I really like when a DM weaves PC backstories into a game. It adds a lot of depth and makes characters more than just a tool to unlock doors and collect treasure.  My friend and her little sister wrote really good backstoreies that have common ellements so I ran with it. Essitially the backbone of the entire heroic campaign was based on them. 

We played our first campaign last night. It went fairly well for four new players and one 3.5 veteran. However the little sister was very frustrated with her character. During the non combat portion she got several bad rolls that resulted in failures. The other characters were able to make up for the bad rolls and the party won the night, but the little sister was still upset with the way things unfolded. I don't feel that my checks were too hard, most could have suceeded on a roll 10 or 11.  

Eventually, she decided that the character she rolled was not the character she wants to play. I already have the house rule that untill they reach level 3 players can swap any skill, power, or feat inbetween games with no penalty. Unfortunatley she wants to do a complete reroll, with new backstory race and everything. She even wants to switch roles (she's the only Leader in the group)

This is going to make a giant plot hole and leave the party unbalanced. What should I do?

First off, yes... absolutly let her change.  I did something similar my first campaign.  Everyone got a reroll at level 2.  Main difference is that i planned it in.  (of course, i managed to kill all but 1 of them anyways about 2/3 of the way though the level...  so yea).

Anyways, lot's of ways you can go...

1) Try and tranfer the story.  Might of been a completely new class, but the same "character".
2) There's compainion rules in the DMG2.  So you can still have the same character go along.
3) Have the character get kidnapped.  Rescue mission is always a good time.
4) Betray the party.  You don't get many opportunities to do that (cause you don't want player vs player).

Remember, if you didn't tell them part of the plot, it didn't happen yet.  It's generally why i don't plan more then 2 sessions at a time.

Also, don't worry if there's no healer.  Just add more potions of healing to their loot.

I'd say take a look at the little sister's character first, and see how well optimized it is in respect to the other characters.  If she's frustrated that her character is feeling weak compared to the others, maybe it is because she is a little bit.  As a DM, I like to help new players build characters that are mechanically decent, if only to keep some balance between my newbies and my char-opping munchkins.

If she's only been around for a session or two, maybe you can keep most of the backstory, but retcon her as a different class and/or race more to her satisfaction?  Or, if it is legitimately just a night where the dice gods didn't smile upon her, maybe she will be luckier next time and appreciate the character more.

Forcing someone to stick with a character they don't want to play isn't going to be fun for either of you.  You may have to change the story around a bit.
 
If role balance is that important, you could be a little more generous with the healing potions and encourage others to multiclass in leader classes so the party at least has some dabblers in leading.
 
Also, I think we have to recognize that not all players like to play the leader, and not all parties are as appreciative of the leader as they should be.  I remember one game where the other members of my party were complaining about my character because I wasn't dishing out enough damage.  Of course, they failed to take into account all the times I healed them.  Also, sometimes it can become a chore to simply be a back-rank healbot for a guy who gets to run up and slash away at the bad guys.

How old is Little Sister? You might want to adjust your expectations of stability and maturity, to avoid a repeat surprise.

But in general, what everyone else has said. Look for a way to say yes.

National Novel Writing Month's website has tips for what to do when your story is stuck. One of them is "Unexpectedly kill a major character". (It's obvious that the Disney Princess and the Dashing Hero now fall in love and get married and that's the end of a rather cliche'd short story... oops, this is supposed to be a novel... the Dashing Hero accidentally dashes in front of an arrow and now the Disney Princess must go back to the old witch ... quickly find another Dashing Hero ... find a farm that needs a pig-herder ... get to the castle on her own ... become a Dashing Hero herself.)

So if you can't proceed without her character, turn it into a Companion Character and look for a chance to kill it without disrupting your story more than you can deal with.

Thanks for all the great tips on how to say Yes.