The basic meaning of the latin word "momentum" is movement or the power to move. The specific use of the term to mean mass $\times$ velocity was a comparatively late (seventeenth century) development.
But "moment", the anglicised version of the word, had already gone into speech to mean the power to get things going in the sense of importance or consequence; Shakespeare's Hamlet talks about "enterprises of great pith and moment". The word is still occasionally used in this sense in everyday speech, and the adjective "momentous" is quite common.
So the turning moment of a force is the force's ability to turn things. Multiplying the force by the perpendicular distance from a point is what gives the force this ability, this consequence!
I suspect that the using the term "moment" when we multiply of a quantity by a significant distance (as when we calculate a dipole moment) is modelled on the calculation of the moment of a force, but the basic idea of giving importance to the quantity by so doing is still in the background.