Gelfand-Naimark Theorem The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that if $ X $ and $ Y $ are compact Hausdorff spaces, then they are homeomorphic iff $ C(X) $ and $ C(Y) $ are isomorphic as rings. Are these two related anyway?

The second theorem you describe is the Banach-Stone theorem. The commutative Gelfand-Naimark theorem says something stronger, namely that every commutative (unital) C*-algebra is of the form $C(X)$ for some compact Hausdorff space $X$. The strongest version of the theorem says that the functor $X \mapsto C(X)$ is a contravariant equivalence of categories. 
I don't know the history here, but both Gelfand-Naimark theorems are "Cayley theorems" for C*-algebras, one saying that commutative C*-algebras can be represented faithfully as function spaces and the other saying that noncommutative C*-algebras can be represented faithfully as spaces of operators. 
