The Death of Ivan Ilych
You might say Vasya's the odd one out in Ivan's family, because he actually seems to care about his father. He's Ivan's only living child besides Lisa. Vasya is a boy of about thirteen or fourteen at the time of Ivan's death. Vasya is still in school, and apparently a source of conflict between his parents; Ivan wanted him to be a lawyer, but Praskovya Fedorovna put him in the high school instead of the law school. According to Peter Ivanovich, Vasya is just like Ivan was when he was younger.
Vasya as the Uncorrupted Child
Though he doesn't get much screen time in the book, both of Vasya's appearances occur when Ivan is sick and are pretty important. The night that Praskovya Fedorovna takes Lisa and Fedor Petrovich to the theater, Vasya goes with them. They visit the ailing Ivan first, and when Vasya comes in, Ivan notices his pathetic, "frightened look of pity" and thinks to himself that "Vasya was the only one besides Gerasim who understood and pitied him" (8.51).
This assessment makes Vasya one of only two people whom Ivan doesn't think is false. Vasya's second appearance is at the very end of the novella, just before Ivan dies. Full of sadness at seeing his father suffering, Vasya takes Ivan's hand and kisses it. It's just at that moment that Ivan sees the light (though we get no clear indication that Vasya caused it). Vasya is the first member of the family for whom Ivan feels compassion.
We get the impression that Vasya can feel more compassion for his father than the others because he's still a child. He hasn't yet been caught up in the poisonous, false society in which all of the adults live. As a result, he doesn't yet have the blindingly selfish focus on his own happiness that the other character seems to have. Vasya can't ignore his father's obvious suffering. Tolstoy suggests through Vasya's character that there's a natural innocence and moral goodness to childhood, which society later corrupts and destroys.
The Vasya / Little Ivan Connection
If all of Ivan's friends are meant to reveal how false Ivan himself was before his illness, Vasya seems to provide us with a link to the innocent child Ivan once was. Tolstoy even reinforces that link by telling us several times how much Vasya is like the young Ivan: he's introduced as "the schoolboy son of Ivan Ilych, who was extremely like his father" (1.48).
We can't forget that it was in Ivan's childhood alone that he felt genuinely happy and not false. Ivan also seemed to have a moral sense then that his later experiences in society negated. Remember those horrid things the narrator tells us he did at one point? Ivan lived in a society that taught him to disregard his view that these really were horrible actions. Instead, he was taught to look at them instead as healthy and normal.
But there's a sign that Vasya's headed in the same direction as his father. The narrator makes one interesting remark about Vasya in the first chapter: "His tear-stained eyes had in them the look that is seen in the eyes of boys of thirteen or fourteen who are not pure-minded" (1.48). What does that mean? We don't get any further details. But most likely it suggests that Vasya's own innocence is already ending. For Vasya, that false world of adulthood (and its horrid things) is just around the corner.