On the most literal level, the dead mouse in Lennie's pocket shows that Lennie is mentally challenged. What sane person would carry a dead mouse in his pocket? In addition to this point, Lennie probably killed the mouse, because he does not know his own strength.
Second, the dead mouse shows that Lennie loves soft things. As the book unfolds, it becomes clearer that Lennie loves all soft things. Later he will love the dog that Slim gave to him. Unfortunately, he will kill it as well as Curley's wife, as he touches her soft hair.
Third, the dead mouse in Lennie's pocket also shows his love for animals. One of the recurrent stories that he wants to hear from George is about the farm they will have in the future. On this farm, Lennie will be charge of the rabbits.
Here is a quotation on this:
“An’ live off the fatta the lan’,” Lennie shouted. “An’ have rabbits. Go on, George! Tell about what we’re gonna have in the garden and about the rabbits in the cages and about the rain in the winter and the stove, and how thick the cream is on the milk like you can hardly cut it. Tell about that, George.” “Why’n’t you do it yourself? You know all of it."
“No . . . . you tell it. It ain’t the same if I tell it. Go on . . . . George. How I get to tend the rabbits."
Finally, on a more figurative level, the dead mouse represents that Lennie's dream will probably be destroyed.