The following answer is entirely speculative (I never knew Leray and even if I did, I doubt we could have had a meaningful exchange on the topic, considering that I was four years old when he died).
I recently had a somewhat vague, yet rather vivid visualization of spectral sequences occur to me, which makes an interpretation of the term along the lines of "spectral lines" in physics quite plausible.
In fact this visualization was induced by the mentioned Timothy Chow article, where he mentions that decomposing the computation of the (co)homology of a graded complex which is compatible with the differential in the sense that $C=\bigoplus G^k C$, $d|_{G^kC}:G^kC\rightarrow G^kC$ is easy, since the (co)homology is the direct sum of the (co)homologies of the graded subobjects.
But on the other hand, if we have a filtration $F^kC\supseteq F^{k+1}C$, which is compatible with the differential in the sense that $d(F^kC)\subseteq F^kC$, then the main complication is that although the differential can never push an element out of the filtration level $F^kC$, it is possible that an element $x=dx^\prime\in F^k C$ that belongs to the image of $d$ is such that $x^\prime\in F^lC$ for $l<k$, so the images of the differentials can come from "below", which is never possible in the graded case.
To fix notation, let $(C,d)$ be a differential vector space (vector space to avoid extension problems) and suppose that $C$ is equipped with a decreasing filtration
$$ F^0C=C\supseteq F^1C\supseteq\cdots\supseteq F^nC\supseteq F^{n+1}C=0, $$ and $d$ is compatible with the filtration.
The whose situation can be visualized in the following diagram:
This should be read as follows:
- The area between $F^p$ and $F^{n+1}$ represents the $p$th filtration level $F^pC$.
- The strip between $F^p$ and $F^{p+1}$ (annotated by $G^p$) corresponds to the $p$th graded subspace $G^pC:=F^pC/F^{p+1}C$ of the associated graded vector space.
- Elements between $F^p$ and $F^{p+1}$ are those which belong to the $p$th filtration level but not to the $p+1$th.
Since $d$ is compatible with the filtration, an arrow may never go down, but they can go up.
If all of the arrows are horizontal, i.e. they never push any element up the filtration level, then $d_1:E_1\rightarrow E_1$ in the spectral sequence is zero, so $E^p_\infty=E^p_1\cong H(G^pC,d_0)$, and $H(C,d)\cong\bigoplus_{p=0}^n H(G^pC,d_0)$, so we recover the case when the vector spaces is graded and the differential is compatible with the grading.
But if there are slanted arrows, then to construct the (co)homology $H(C,d)$, we need to take more quotients. An element is a cycle if the differential vanishes on it, which in the present diagram means that the differential pushes it above $F^{n+1}$ (only the zero lives there), and to construct cycles, we can first construct the associated graded and check that the differential there is zero (so $d$ is not horizontal), then we can go to $E_1$ and check that the differential there is also zero (then the differential goes up at least 2 filtration levels), then we go to $E_2$ etc. and verify that the differential is zero in each subquotient until we exhaust the filtration.
Likewise, to construct boundaries, we first check that the element is the target of a horizontal arrow (then it is a boundary in $E_0$), if not then we check that it is a target of an arrow that goes up 1 (then it is a boundary in $E_1$) and if not then we check that it is the target of an arrow that goes up 2, and so on. If we manage to exhaust all filtration levels without finding an arrow whose target is the element, then it is a nontrivial cohomology class in the original complex.
Of course this is not new to most people here and I don't intend to lecture anyone on spectral sequences (nor am I an expert on the topic, really). But coupled with the diagram above, this is rather reminiscent of e.g. the spectral lines of an atom.
The slantedness of the arrows are like transitions between energy levels. To compute the (co)homology, we need to understand transitions to the ground state ($F^{n+1}=0$), and if it is too hard to understand them directly, we decompose them into transitions between neighbouring lines, understand those, and put together the full picture from these partial transitions. Essentially we spectrally decompose a (co)homology problem.
Of course this is just an analogy between visualizations with no real relationship between the two concepts ((co)homology computations and spectra of atoms), but the visual analogy is rather striking I think.
And of course I have no way of knowing whether Leray really had an image like this in mind, but I find it plausible. Especially that of course at the time this was not a well-developed theory with the usual syntax and ready-made proofs, and spectral sequences are rather complicated, so I would be very surprised if he had no such intuitive picture during its development.
