The Wikipedia quote appears to be lifted from this Solid State Physics text by CTI Reviews, and then plastered all over the web. The text does not give any citation of Lanczos, however. Here is the only passage in Lanczos's 300 pages long Variational Principles of Mechanics that contains the word "self-adjoint":
"Schroedinger, on the other hand, introduced the operational viewpoint and reinterpreted the partial differential equation of Hamilton-Jacobi as a wave equation. His starting point is the optico-mechanical analogy of Hamilton. In spite of the radical departure of the new concepts from those of the older physics, the basic feature of the differential equations of wave-mechanics is their self-adjoint character, which means that they are derivable from a variational principle. Hence, in spite of all differences in the interpretation, the variational principles of mechanics continue to hold their ground in the description. In spite of the radical departure of the new concepts from those of the older physics, the basic feature of the differential equations of wave-mechanics is their self-adjoint character, which means that they are derivable from a variational principle. Hence, in spite of all differences in the interpretation, the variational principles of mechanics continue to hold their ground in the description of all the phenomena of nature." [emphasis mine]
So Lanczos only claims the converse, that self-adjointness implies a variational formulation. This is essentially right. Of course, if one wants a variational principle to involve actual optimization rather than just extremals the spectrum also needs to be bounded from below, but that is usually the case in "physical" problems. If $A$ is the operator the variational functional will then be $f(x)=(Ax,x)$ with some constraints on $x$, e.g. normalization and/or boundary conditions.
What the "quote" ascribes to Lanczos, on the other hand, is nonsensical: a variational principle which is non-linear (more precisely, whose functional is not quadratic), or not stated in a Hilbert space does not "describe an expression which is self-adjoint or Hermitian". There is even more confusion in the expanded version of the "quote" in the Schools Wikipedia History of Physics:
"According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint or Hermitian. Thus such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. Noether's theorem identified the conditions under which the Poincaré group of transformations (what is now called a gauge group) for general relativity define conservation laws. The relationship of these invariants (the symmetries under a group of transformations) and what are now called conserved currents, depends on a variational principle, or action principle. Noether's papers made the requirements for the conservation laws precise. Noether's theorem remains right in line with current developments in physics to this day."
Aside from the additional mess with "Hermitian transformation" and conservation laws, this possibly refers to Appendix II or section 20 of chapter XI ("Noether’s principle") in the fourth edition of Lanczos’s book. There he determines the conservation laws associated with the invariance of the Lagrangian under a phase shift for Maxwell’s and Schrödinger's equations, see also The Noether Theorems by Kosmann-Schwarzbach.