Why are spectral sequences called "spectral"? Is that use of "spectral" related to other uses in math, such as spectra in homotopy theory, the spectrum of a ring in algebraic geometry or the spectrum of an operator? Why are those things called spectral, for that matter?
As per Adrián Barquero's request, I post my comment as an answer (even if I think that it doesn't answer the question but rather indicates that the answer isn't really known).
See the related discussion on Math Overflow.