It is a quotient: a financial aggregate representing the value of transactions in a given time period divided by a financial aggregate representing the amount of the money available. Economists have particular problems defining the money supply.
So the _velocity of money_ is an average over a time period and has dimension $T^{-1}$.
As far as I am aware, it is not the derivative of anything and, even if the concept of the instantaneous velocity of money was theoretically possible by taking arbitrarily shorter time intervals, its integral would simply be the average using a different unit of time.
This suggests _velocity of money_ is an example of a scientific and mathematical concept abused by economists.
So _speed of money use_ might be slightly better as there is no direction, and _frequency of money use_ could be even closer to the concept as there is no distance.