I might be wrong about the exact historical reason. But the way I see it "spectral decomposition of A" is a way to express A as a sum of orthogonal, rank-1, op

I might be wrong about the exact historical reason. But the way I see it "spectral decomposition of A" is a way to express A as a sum of orthogonal, rank-1, operators. A = \sum l_i u_i u_i^T. Those l_i are the eigenvalues; u_i are the eigenvectors. The eigenvectors look a whole lot like the "modes" in a Fourier decomposition. And if you plot (i, l_i), the eigenvalues are a bit like the "spectrum" (the amplitude of each mode). In fact, the complex exponentials (the modes in the Fourier decomposition) are also eigenvectors of a specific operator (the Laplacian). Math people are good at finding connections between things.