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 we have the adjective "momentous".
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, indeed this moment!
I suspect that using the term "moment" when we multiply of a quantity by a significant distance, as when we calculate an electric dipole moment as $Q\Delta \vec r$, is modelled on the calculation of the moment of a force, but the basic idea of giving importance to the quantity (in a particular context) by so doing is still in the background.
And what about the moment of inertia of a rigid body? To calculate this we multiply each mass element by a distance squared, rather than by a distance, before summing. But again we are constructing from the masses a quantity of moment when it comes to rotation.
This answer (especially the last paragraph) probably seems rather hand-wavy, but don't forget that it is only names that we are talking about. We could call the moment of inertia "Charlie" and it wouldn't affect the Physics. Names of quantities in Physics are usually very logical and helpful, but not always, as in the case of electromotive force. Moment is not quite in this category, but it's possibly not the most transparent of terms.