One way to think about momentum is to make an analogy with static equilibrium.
Let two weights, m1 and m2 be on the ends of a seesaw, the seesaw is such that you can move the position of the fulcrum.
The seesaw is in static equilibrium when the fulcrum is located at the Common Center of Mass (CCM) of m1 and m2. You have radial distance r1 and r2. Then we have for the static equilibrium:
m1r1=m2r2
Now take two masses m1 and m2, and use the CCM of m1 and m2 as the zero point of the coordinate system that you use to represent the respective velocities of m1 and m2 .
Let the initial situation be that the two masses are moving towards each other, so there will be a collision. Let that be a perfectly elastic collision.
Among the first to examine that kind of collision was Huygens. He set up two pendulum bobs to collide with each other. By having the bobs hit each other dead center Huygens obtained fairly good elastic collisions. Huygens had no way of making accurate velocity measurements, but he could measure how high bobs swing back up after they have collided.
Huygens' experiments corroborated the following expectation:
If the bobs swing towards each other with their velocities in the following ratio:
m1v1=m2v2
Then after the elastic collision the velocities are reversed, and still in the same ratio:
m1v1=m2v2
The supposition then is that the quantity mv expresses something that we can refer to as 'quantity of motion'.
The supposition is that two objects of different mass bring the same amount of quantity of motion to the CCM by virtue of the quantity mv being equal.
The clincher is that change of momentum is galilean invariant; you can use any inertial coordinate system to describe the collision; the amount of change of momentum is independent of the choice of inertial coordinate system.