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Dill's motivation for lying is that his fantasy world is much better than the real one in which he lives. Dill is a common young man suffering from neglect, and although he has everything he needs to grow physically, he does not get much attention at home. He is sent off in the summers to live with his aunt by a mother who does not show much interest in him. He lies for two reasons: 1) he wants a better life than he believes he has; and 2) he wants attention. He concocts stories to have others listen to him and show interest in him.
When Dill runs away from home and hides under Scout's bed, this truth about him becomes clear. He first lies and says that his new step-father abuses him. If this lie is true, then Dill can stay in Maycomb, where he lives a better life and gets more attention. Also, telling this story gets him both attention and concern from the adults.
However, Dill finally admits that this story isn't true. He was simply angry that his mother seemed to have no time for him, and ran away to Maycomb in anger and looking for friends.
Dill, in part, shows that sometimes things just aren't the way that they seem. Some homes aren't places where children want to be. Jem and Scout see their home as nearly idyllic, even though Scout has some questions about the way Atticus does things. Initially, she can't figure out why Dill would run away from home.