I believe that as sigfpe points out in the comments, the etymology comes from the sort of integrals which appear when one computes the periods of oscillation of mechanical systems.
Indeed, consider a particle of unit mass moving in the real line under the influence of a potential $V(x)$. The phase space of this system is the cotangent bundle of the real line, which we can identify with $\mathbb{R}^2$. The phase space is foliated by the physical trajectories, which in this simple example are labelled by the energy $E$ of the trajectory. Suppose that $E$ is such that trajectories are closed. The following picture illustrates the situation.
Potential function http://dl.dropbox.com/u/5096148/Potential.png
If the particle lies in the interval $[a,b]$ it will remain in that interval for all time and its motion will be periodic with period given by the (improper) integral $$ T = \sqrt{2} \int_a^b \frac{dx}{\sqrt{E-V(x)}}.$$
Here the cycle is the one-dimensional submanifold of the phase space (with coordinates $(x,y)$) given by the equation $$\frac12 y^2 = E - V(x),$$ and the period is the integral of the differential $dx/y$ on the cycle.
A typical example is that of a simple pendulum, where $V(x) = g\ell (1-\cos x)$, where $\ell$ is the length of the pendulum and $g$ the acceleration due to gravity. If we let $E = g\ell (1-\cos x_0)$ then the period of oscillation becomes $$ T = 2 \sqrt{\frac{2\ell}{g}} \int_0^{x_0} \frac{dx}{\sqrt{\cos x - \cos x_0}}.$$
To turn this into an elliptic integral we change variables to $\theta$ defined by $$ \sin\theta = \frac{\sin x/2}{\sin x_0/2} $$ in terms of which the period integral becomes an elliptic integral of the first kind $$ T = 4 \sqrt{\frac{\ell}{g}} \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-\sin^2(x_0/2)\sin^2\theta}}.$$
I'm not sure of dates, but I would be surprised if this (which was certainly known to the Bernoullis) did not predate the uses of period in algebraic geometry.