Before Newton's time, it was thought that an object only moved while a force was continuously applied. Newton invented the concept of "momentum" meaning the tendency of a moving object to continue moving in the absence of any applied force. At the time Newton published the Principia, impetus was the quality of an object that was moving independent of an observed force. Impetus comes from the Latin in- + petere to go to, seek -- from Greek petesthai to fly, piptein to fall, pteron wing. Also, push and pull derive from the Latin pellere.
However, the equation p=mv wasn't given first by Newton, but was developed afterwards. Many scientist/mathematicians developed what we now call "Newtonian Mechanics," and it's easy to imagine some sticking with the old impetus while others used the new momentum. P was a convenient symbol -- m would be confused with mass, i is too often used to indicate an instance of an object. (Mi might be confused with mi, which usually means the mass of the ith object.)