Not many answers, huh?
Lisa can get very angry only because of her beliefs. And I almost share those with her, so I'll explain how I'd behave if Lisa was a little angry at me:
I'd honestly ask her of the exact reason of her anger, and if she couldn't explain it I'd tell her to calm down, maybe play the sax, until she finds the right words. Because I'm ready to accept the opinions of others, and she'd understand since she has similar traits. And I think she'd then forgive me.

I've needed 15minutes for that text cause I really tried to imagine such a situation. Oh now it's re-submit 

SimpReal
Hm.. can't remember, but I do know Gustavo's post is gone! =(

I'm gonna go with Maléfix on this one for most of it.
People get mad at me approximately.. never. And having a friend as precious as Lisa, I wouldn't even dream of disappointing her. If I'd for some reason ever let her down, there would surely be a good explanation, and one way or another I think she'd understand. Homer's got away with a few things without good explanations, right? ;x
Now... let's forget everything I just said. If she would get really upset with me, I'd of course be sad (whether she understands it or not) to have disappointed her. If there'd be absolutely no way to make things up to her I'd be devastated. Although I can't imagine a situation where I'd "realise" that. o_O I'd keep trying to set things straight in any way I can, while trying not to bother her by doing so. It's difficult to say what I'd do if I gave up, depends on what my life looks like overall at that point. At the moment I don't have anything to cling on to, so I'd probably go depressed and ignore my life for at least a few months. If things were good I'd probably make sure to keep earning my grades at school, do my chores and what not, but still be pretty sad for most part of the days. If things were really good and I also had some other friends that I enjoy talking to, I'd probably be sad at times, but still be able to enjoy daily life.

I realise perfectly that with my endless laziness and ignorance I would be a PITA for Lisa. If we were near to each other, I can easily imagine situation when I disappoint her, even close to the bound of breaking. What would happen afterwards? Okay it would make me very depressed, but, I wouldn't kill myself for sure, I wouldn't savour my depression either. I think I would wait few days and try and talk the problem over, to say sorry if needed. Lisa is a nice person, she is quite able to forget and to forgive, and time cures everything. But if, like Shadow Nait said, she would never forgive you... damn it would be awful. The best way would be to move somewhere away and to try and live without hurting others.

Maybe in several months she would find you and try to make things up... that sounds like a dream and therefore unlikely, but if that happens, don't miss your last chance. :-)

Lisa?? Lisa Simpson? Disappointed in me? The universe would collapse if that happen. Because it never happen. I disappoint nobody who I care for. All people know that. My family. My girlfriends. My friends. Even my former girlfriends.

Well, duh... it would be same as the situation any other important person to me would be disappointed in me. I would feel very upset with myselfe. Same with my girlfriend. If she is for some important reason dissappointed I try everything to make her happy again.

oh that's true of course. But I think I'd mostly do what I already posted, if I wasn't already in a time-space-collapsed dimension...

Okay... I'll indulge in a little bit of hypothetical thinking here...

Assuming Lisa was real, and assuming she was a friend of mine, if she were to be upset at me somehow, I'd pretty much treat it like I do with everyone else. For example, a friend of mine gets mad at me, and I don't know why. I ask him or her what the problem is, and either they tell me and we work things out (usually), or they refuse to speak to me. At which point I just ride it out until they come around, which can be quickly when they realize that the silent treatment goes nowhere in solving problems.

Gary made a good point though - it really depends on why she's mad at you. I've had friends mad at me for a variety of reasons, some of them are my fault, some of them are their fault, some are just plain silly. So it's entirely circumstantial.

Miles
Well, Lisa is just as human as the rest of us. If I recall in one episode last season she was upset at Homer because he was refereeing one of her soccer games, and made a fair call that was in favor of the other team. After this she became upset with him and wouldn't talk to him because he "ruined her life".

My point is, she's not perfect, plus she's eight years old. There's much plausible cause for her to be angry at someone, and for it to be her own fault.