In addition to the historical component, the question also asked about the relation between spectral theory and Stone spaces.
$\def\Z{{\bf Z}} \def\R{{\bf R}} \def\C{{\bf C}} \def\Spec{\mathop{\rm Spec}}$

Given an algebra-like object $A$, we assign to it its poset of ideals
(typically defined as kernels of homomorphisms $A→B$),
which is interpreted as the poset of opens of some space $S$.

The technical term for such posets is [locale](https://ncatlab.org/nlab/show/locale),
which is a notion very closely related to topological spaces.
In particular, from any locale one can canonically [extract a topological space](https://ncatlab.org/nlab/show/locale#RelationToTopologicalSpaces),
and this is the topological space $S$ produced in many classical Stone-type dualities.
The points of $S$ are ideals corresponding to morphisms $A→k$, where $k$ is often a particularly simple algebra.
These often turn out to be maximal ideals in $A$.

Conversely, given a space-like object $S$, we assign to it the algebra of morphisms $S→k$,
where $k$ is often the “same” algebra $k$ as above,
only this time its underlying object is a space, not just a set.

Some examples from general topology, measure theory, differential geometry, algebraic geometry, and complex geometry
(the list is very much incomplete):

|algebra|homomorphism|$k$|ideal|space|maps|
|-|-|-|-|-|-|
|Boolean algebra|homomorphism|$\Z/2$|ideal|compact totally disconnected Hausdorff|continuous map|
|complete Boolean algebra|complete homomorphism|$\Z/2$|closed ideal|compact extremally disconnected Hausdorff|open continuous map|
|localizable Boolean algebra|complete homomorphism|$\Z/2$|closed ideal|hyperstonean space|open continuous map|
|localizable Boolean algebra|complete homomorphism|$\Z/2$|closed ideal|[compact strictly localizable enhanced measurable space](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical/20820#20820)|measurable map|
|commutative von Neumann algebra|normal \*-homomorphism|$\C$|closed \*-ideal|compact strictly localizable enhanced measurable space|measurable map|
|commutative unital C\*-algebra|\*-homomorphism|$\C$|closed \*-ideal|compact Hausdorff space|continuous map|
|commutative algebra over $k$|homomorphism|$k$|ideal|coherent space / affine scheme|continuous map / morphism of schemes|
|finitely generated germ-determined C$^∞$-ring|C$^∞$-homomorphism|$\R$|germ-determined ideal|smooth locus (e.g., smooth manifold)|smooth map|
|finitely presented complex EFC-algebra|EFC-homomorphism|$\C$|ideal|globally finitely presented Stein space|holomorphic map|

The duality relevant to the spectral theory is the duality between commutative von Neumann algebras
and [compact strictly localizable enhanced measurable spaces](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical/20820#20820).

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.)

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 $\C$.

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.).

Given a vector bundle-like object $V→S$, we assign to it its module of sections,
which is a module over the algebra of maps $S→k$.
Conversely, given a module $M$ over $A$,
the corresponding vector bundle-like object $V→S$ over $S=\Spec A$
has as its fiber over some point $s∈S$ the vector space $M/IM$,
where $I$ is the ideal corresponding to $s$.
(Many details are necessarily omitted in this brief sketch.)

Typically, genuine vector bundles correspond to dualizable modules
(dualizable with respect to the tensor product over $A$).
Non-dualizable module tend to correspond to sheaves that are not vector bundles,
e.g., skyscraper sheaves etc.

|module|vector-bundle-like object|
|-|-|
|module over a Boolean algebra|sheaf of $\Z/2$-vector spaces|
|Hilbert W\*-module over a commutative von Neumann algebra|measurable field of Hilbert spaces|
|representations of a commutative von Neumann algebra on a Hilbert space|measurable field of Hilbert spaces|
|Hilbert C\*-module over a commutative unital C\*-algebra|continuous field of Hilbert spaces|
|module over a commutative algebra over $k$|sheaf of modules over an affine scheme|
|dualizable module over a commutative algebra over $k$|algebraic vector bundle|
|dualizable module over a finitely generated germ-determined C$^∞$-ring|smooth vector bundle|
|dualizable module over finitely presented complex EFC-algebra|holomorphic vector bundle|

The duality relevant to the spectral theory is the duality between representations of a commutative von Neumann algebras on a Hilbert space
and measurable fields of Hilbert spaces.

Given a normal operator $T$ on a Hilbert space $H$,
$T$ generates a commutative von Neumann algebra $A$ inside $B(H)$,
whose spectrum $\Spec A$ is a compact strictly localizable enhanced measurable space.

Furthermore, the inclusion of $A$ into $B(H)$ is a representation of $A$ on $H$.
As such, it corresponds under the Serre–Swan-type duality to a measurable field of Hilbert spaces over $A$.
This is precisely the measurable field produced by the classical spectral theorem.

Under the duality, the operator $T$ corresponds to the operator that
multiplies a given section of this measurable field of Hilbert spaces
by the complex-valued function $\Spec A→\C$ produced above. 
Thus, we recovered the entire content of the classical spectral theorem.

In fact, the above considerations work equally well to establish the spectral
theorem for an arbitrary family (not necessarily finite) of commuting normal operators.