I'm taking a vector analysis class, and upon reaching the Stokes's theorem, our professor briefly mentioned Stokes did not actually prove his theorem. It's funny, but definitely one of many situations where we name theorems after people just "involved" in the theorem but not actually the provers.
I found this very nice paper1 describing the history we know. In short, Ostrogradsky proved the first version of what we know as the divergence theorem. Green then proved a different version of it, which when reduced to two dimensions, derives what we know as Green's theorem (except that Green never seems to have done that).
The first version of Stokes' theorem in $\R^3$ was actually in the Smith's Prize exam from 1854 (yes, it was the question to an exam, specifically question #8 found here2), and the one who and the author of the exam was Stokes himself. So we only call the theorem after him for perhaps shedding light on this problem, but from what we know, the first to actually prove it was Hermann Hankel in 1861, using the results of Green.
Nowadays, we know the divergence theorem, Green's theorem, and the one proven by Hankel are all particular cases of what we know today as Stokes' theorem, involving a generalized manifold with border and exterior derivatives. The first one to combine them all like that was Vito Volterra in 1899, then Poincaré simplified the notation, and then Elie Cartan was the one to introduce the notion of differentiable forms, but only around 1936 he stated Stokes' theorem using the notation we know today. $$\int_M d\omega = \int_{\partial M} \omega$$
1: The History of Stokes' Theorem, Victor J. Katz, Mathematics Magazine, Vol. 52, No. 3 (May, 1979), pp. 146-156
2: Smith’s Prize Exam. February, 1854. PDF provided by clerkmaxwellfoundation.org