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

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,
which is a notion very closely related to topological spaces.
In particular, from any locale one can canonically extract a topological space,
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 |
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.

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.).

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.

See also the nLab article duality between algebra and geometry, which may contain additional updates.

Stone spaceshas some useful historical information. $\endgroup$ – Noah Schweber Apr 13 at 12:48