Personally, I don't consider the Stone Representation Theorem and the GNS-construction to be directly related. However, the former is closely related to the Gelfand representation, which in a way is the commutative version of the Gelfand-Naimark theorem. (Yes, a lot of theorems in the study of Banach algebras are named after Gelfand.) The proofs for Stone representation and Gelfand representation are virtually identical:
- Consider the set of maximal ideals.
- Note that every maximal ideal gives the same quotient ring. In the case of commutative $C^*$-algebras the quotient is always $\mathbb C$, in the case of Boolean rings the quotient is always $\mathbb F_2$, the field of two elements. (In a general commutative ring $R$, different maximal ideals $\mathfrak m_1,\mathfrak m_2 \subseteq R$ typically give rise to quotients $R / \mathfrak m_1$ and $R / \mathfrak m_2$ that are not isomorphic. For instance, every finite field $\mathbb F_p$ of prime order occurs as a quotient of the ring $\mathbb Z$ of integers.)
- Define some suitable topology on the maximal ideal space.
- This way we obtain a duality between seemingly unrelated topics in mathematics (algebraic objects versus certain topological spaces). Stone gives a duality between Boolean algebras and Stone spaces; Gelfand gives a duality between commutative $C^*$-algebras and locally compact Hausdorff spaces.
Perhaps the GNS-construction is then a non-commutative analogue of Stone duality. However, because of its non-commutative nature, the proof is much more difficult, and loses most of the resemblance to the Stone Representation Theorem. Moreover, it doesn't give you any kind of duality in the way that the Stone and Gelfand representations do.
Furthermore, I stumbled upon this question on MathOverflow about the influence of the Gelfand representation in other areas of mathematics. I think you will find this very interesting. According to the OP:
Another related result is Stone's representation theorem from 1936, and a good summary of this circle of ideas can be found in Varadarajan's Euler book.
Seems like we have to check out that book! :D