The concept of torque, also called moment of a force, as you've pointed out correctly, originated with the studies of Archimedes on levers. The term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884, as suggested here.
The Mathematical form of Torque as we know it now is $\vec{\tau} = \vec{r} \times \vec{F}$. Or $\tau = r \cdot F \cdot \sin{\theta}$
While studying/using levers the Law Of The Lever was used which just gave us the mechanical advantage of the lever.
Take this scenario:
If a and b are distances from the fulcrum to points A and B and the force FA applied to A is the input and the force FB applied at B is the output, the ratio of the velocities of points A and B is given by $\frac{a}{b}$, so we have the ratio of the output force to the input force, or mechanical advantage, is given by $\frac{F_B}{F_A} = \frac{a}{b}.$
Archimedes proved it using Geometric Reasoning, as stated
here.
After which Newton's Law's helped grasp rotational dynamics better.
To Learn more about Torque. Also check Wikipedia.