Because the result of multiplying m times v is a very useful quantity in physics, and can be used to help understand how particles (and other objects) behave when acted upon by forces, and when interacting with each other.
It can be shown to be a conserved quantity in an isolated system from Newton's Second Law. Which is really important for physics. And in modern physics it has an even broader application because it can include variations in mass. p = momentum
[tex]\sum F = ma = m\frac{d\textbf{v}}{dt} = \frac{dm\textbf{v}}{dt} = \frac{d\textbf{p}}{dt}[/tex]
So an isolated system ( no net force ) has a conserved momentum or the time derivative of the product of mass and velocity is equal to zero.