The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; then the prime spectrum (which is equal to the maximal spectrum) is precisely the set of eigenvalues of $T$. The use of the term "spectrum" in the operator sense, in turn, seems to have originated with Hilbert, and was apparently not inspired by the connection to atomic spectra. (This appears to have been a coincidence.)
A cursory Google search indicates that Hilbert may have been inspired by the significance of the eigenvalues of Laplacians, but I don't understand what this has to do with non-mathematical uses of the word "spectrum." Does anyone know the full story here?
Best Answer
Hilbert, in fact, got the term from Wilhelm Wirtinger (the first one to propose it according to, say http://www.mathphysics.com/opthy/OpHistory.html)
the paper of Wirtinger is "Beiträge zu Riemann’s Integrationsmethode für hyperbolische Differentialgleichungen, und deren Anwendungen auf Schwingungsprobleme" (1897).
In http://jeff560.tripod.com/s.html it says
"...Wirtinger drew upon the similarity with the optical spectra of molecules when he used the term "Bandenspectrum" with reference to Hill’s (differential) equation."
I haven't read Wirtinger's paper, nor do I know how reliable these sources are :)