Lisa represents all of the things the other characters in the show do not not: she is well read, gets outstanding grades, has tremendous musical talent, has a concern for the well-being of others and is just generally a good kid. However, Lisa has a know-it-all attitude which separates her from peers becasue she is the student all of teachers and parents love. She feels more at home with members of MENSA in Springfield than she does with her own family. Lisa is truly happy when she has someone with whom she can share her talents. In one case, after Homer had a crayon removed from his brain that has been stuck for nearly all of his adult life, his IQ went up dramatically and the two became the best of friends.
Almost every family has a child that has some special skills that the others do not, wether it be athletically or musically. For this reason it is easy for everyone who watches the show to identify with the perfect-child stereotype becasue they either have one in their family or have had experience with a brainiac when they went to school. Lisa represents both sides of the perfect-child, the one with all the gifts but also the one who everyone, at least under the surface, dislikes due to all her talents. Because every person who watches the show has spent time around a person like Lisa it makes the show more watchable.