A critical bit of the basic understanding of object references is their scope or life span. The temptation to set a variable to null to indicate we're done with it tells us something about the code. If part 1 of a method uses some object and then sets it to null when it's done, and part 2 of the method doesn't use that object any more, then the method is probably doing two distinct things that could be broken out.
Now any local variables in partA will go out of scope when it returns. The object reference won't be visible in partB making it easier to read and reducing the chances of a silly mistake from trying to use it again.
Making something eligible for GC doesn't generally influence my thinking here. But reducing the scope to expose a variable only to code that really needs it is often on my mind. Earlier GC eligibility may be a happy side effect.
BTW: I showed separate methods as a way to control variable scope. As you read up on variable scope, find others.