In an $R$-module $M$, an element $m \in M$ is said to be torsion if $am = 0$ for some $a \in R$ with $a \neq 0$.
Also, for a non-orientable (closed) surface such as the projective plane or the Klein bottle, there is a $\mathbb Z/2$ term in the first homology. This part is said to detect the "twisting" in the surface.
So this leads to the plausible notion that the origin of the word "torsion" in algebra is related to this "torsion" or "twisting" from topology.
Unfortunately the difficulty is that this is not exactly true. For example, the Möbius band is surely a twisted object. But it deformation retracts to the circle, and therefore its homology is very normal.
I hope somebody can shed more light on this terminology.