Steinbeck had to work a mouse into his novella because he takes its title from a poem by Robert Burns called "To a Mouse" (translated):
But Mouse, you are not alone,
In proving foresight may be vain:
The best laid schemes of mice and men
Go often askew,
And leaves us nothing but grief and pain,
For promised joy!
Or did he title the novel after he wrote it? Hmm...
Lennie is fixated with mice because they are easily petted. They're small and can fit in his pocket. He can hide them from Georege (who won't let him have any animals--because he kills them by petting them too hard). And they're easily disposed of. And he won't get into too much trouble for killing them (they're expendable).
I'm not exactly sure how a big, lumbering, slow-witted man like Lennie could catch a mouse in the first place...
The relationship of the animals that Lennie kills is important. First, he kills a mouse, then a dog, and then a human (Curley's Wife). Lennie moves up the evolutionary ladder. Now, many of us do not think it is a crime or a sin to kill a mouse. But, we do think it is a crime and sin to kill a dog (just ask Michael Vick). Certainly, it is a crime and a sin to kill a human. All of this begs the question: what separates us from animals? Why do we value some life more than others? Can we justify killing some things and not others? Some humans and not others?
Steinbeck looked at human evolution from a Darwinian and Pragmatist point of view. He saw men as animals--at least they were treating each other like animals in the early 20th century (World War I, World War II, Great Depression). It was the worst time in human history. Mass murderers (Hitler, Stalin) and weapons of mass destruction filled the air with gas. So, the novella is a kind of a re-examination of the value of life--even down to the life of a mouse, whose death, of course, foreshadows Lennie's own.