In addition to the historical component, the question also asked about the relation between spectral theory and Stone spaces. $\def\Z{{\bf Z}} \def\R{{\bf R}} \def\C{{\bf C}} \def\Spec{\mathop{\rm Spec}}$$\def\Z{{\bf Z}} \def\R{{\bf R}} \def\C{{\bf C}} \def\Spec{\mathop{\rm Spec}} \def\Proj{\mathop{\rm Proj}} \def\Cont{{\rm C}} \def\MSpec{\mathop{\rm MSpec}} \def\Li{{\rm L}^∞}$
The duality relevant to the spectral theory is the dualityduality between commutative von Neumann algebras and compact strictly localizable enhanced measurable spaces.
Given a normal operator $T$ on a Hilbert space $H$, $T$ generates a commutative von Neumann algebra $A$ inside $B(H)$, i.e., bounded operators on $H$. (This is precisely the point where normality is crucial; without the relation $T^*T=TT^*$ the algebra generated by $T$ will be noncommutative.)
More concretely, $A$ can be described as the closure in the ultraweak topology on $B(H)$ of the set of polynomials with complex coefficients in variables $T$ and $T^*$, i.e., finite sums $∑_{i,j}a_{i,j}T^i(T^*)^j$ for arbitrary $a_{i,j}∈{\bf C}$.
By the cited duality, the commutative von Neumann algebra $A$ is dual to a compact strictly localizable enhanced measurable space $\Spec A$. This is indeed the spectrum of $T$ in the usual sense. Under this equivalence, the element $T∈A$ corresponds to the measurable map $\Spec A→\C$ given by inclusion of $\Spec A$ into $\C$.
More concretely, we can extract all projections from $A$ (defined as self-adjoint idempotents, $T^2=T$ and $T^*=T$), and these form a complete Boolean algebra $\Proj A$. The easiest way to see this is to observe that Boolean algebras are precisely rings in which $x^2=x$. The ring identities are trivially satisfied by definition of projections. Completeness is implied by the fact that $A$ is closed in the ultraweak topology.
The Stone duality converts the Boolean algebra $\Proj A$ into a topological space, its Stone spectrum $\Spec A$. Points in the Stone spectrum are precisely the maximal ideals $P$ in the Boolean algebra $\Proj A$. Open sets correspond to ideals $I$ of $A$: a point $P$ belongs to the open set corresponding to $I$ if $P$ does not contain $I$ as an ideal. In fact, the topological space $\Spec A$ also coincides with the Gelfand spectrum of $A$, interpreted as a commutative unital C-algebra; so points in $\Spec A$ can also be interpreted as maximal closed -ideals $I$ in $A$ as a commutative C-algebra, or, equivalently, as homomorphisms of unital C-algebras $A→\C$.
Furthermore, we have a canonical isomorphism of von Neumann algebras $$R\colon A→\Cont(\Spec A,\C),$$ where $\Cont(\Spec A,\C)$ denotes the algebra of continuous complex-valued functions on $\Spec A$. Concretely, given an element $a∈A$ and a point $p∈\Spec A$ given by the homomorphism $A→\C$ of C*-algebras, we set $T(a)(p)=p(a)$. This is the Gelfand transform of $a$.
The topological space $\Spec A$ belongs to a very special class of topological spaces, the hyperstonean spaces. From such a space one can extract a σ-algebra $M$ of measurable sets and a σ-ideal $N$ of negligible sets (alias sets of measure 0), both on the set $\Spec A$. The resulting triple $\MSpec A=(\Spec A,M,N)$ is an example of an enhanced measurable space. The elements of $N$ are precisely the nowhere dense subsets of $\Spec A$, which coincides with meager subsets (alias sets of first category). The elements of $M$ are precisely the symmetric differences of clopen (closed and open) subsets of $\Spec A$ and elements of $N$. (This is sometimes referred to as the Loomis–Sikorski construction for $\Spec A$; see John C. Oxtoby's book Measure and Category for more on this topic.)
Continuing the above line of reasoning, we have a canonical isomorphism of von Neumann algebras $$S\colon A→\Li(\MSpec A,\C),$$ where $\Li(\MSpec A,\C)$ denotes the algebra of equivalence classes of complex-valued measurable functions $\MSpec A→\C$ modulo the equivalence relation of equality on a conegligible set (alias equality almost everywhere). (Indeed, one can prove that $\Cont(\Spec A,\C)$ is canonically isomorphic to $\Li(\MSpec A,\C)$.)
Under this correspondence, the original operator $T∈B(H)$ corresponds to an element $S(T)∈\Li(\MSpec A,\C)$. The map $R(T)\colon \Spec A→\C$ allows us to interpret ponts of $\Spec A$ (and therefore also of $\MSpec A$) as complex numbers. This identifies $\MSpec A$ with the usual spectrum of the operator $T∈B(H)$. Furthermore, the isomorphism $$S\colon A→\Li(\MSpec A,\C),$$ is known as the Borel functional calculus of $T∈B(H)$. More precisely, given a bounded Borel-measurable function $f\colon \C\to\C$, the element $S^{-1}(f\circ R(T))∈A⊂B(H)$ is precisely the operator $f(T)∈B(H)$ given by the traditional Borel functional calculus.
One may ask whether we can recover the full spectral theorem for a normal operator in this manner. This is possible once Stone duality is upgraded to Serre–Swan-type duality between modules and vector bundle-like objects (including, e.g., sheaves etc.).
In fact, the above considerations work equally well to establish the spectral theorem for an arbitrary family (not necessarily finite) of commuting normal operators.
See also the nLab article duality between algebra and geometry, which may contain additional updates.