In our eyes, there are two ways to look at the title of this sucker.
First, we can think of Me Before You as "what I was before I met you." That's what the book is about to a large degree: the different person Lou becomes after she meets Will.
But we can also think of it as "putting myself before you." Will certainly does this, choosing to go through with assisted suicide despite his growing relationship with Lou. We can also see this dynamic to a lesser degree with Lou, as she slowly comes to put her own needs and desires before those of other people, at least when it is right to do so.
There's also the idea that one of the characters is heading off to the other side before the other one does—but oh dear, we're going to need some more Kleenex now…