What is a diabolical point? A lot of papers define a 'diabolical point' as a "double semi-simple eigenvalue." I know a semi-simple eigenvalue is one which has algebraic multiplicity and geometric multiplicity to be equal. However, I could not find any definition of a double semi-simple eigenvalue.
 A: "Double" simply means a degenerate eigenvalue (repeated root of the characteristic equation), thus a "double semi-simple eigenvalue" is a once repeated eigenvalue (i.e., with algebraic multiplicity 2) that spans a 2D vector space (i.e., its geometric multiplicity is also 2).
You can check, e.g., the second section of this paper (e-print), or 4.1 of this (e-print), or section 9.2.4 of this book, or, apparently, chapter 5 of this book.
These points are relevant mostly because they are associated to systems at bifurcations, i.e., structurally unstable systems whose behavior can change qualitatively under small perturbations. Such sensitivity has been used in the construction of very sensitive sensors, and even more sensitive than the diabolical points are the even more degenerate "exceptional points" (where "not only do resonant frequencies coincide but their resonant modes do too") . Both situations are schematically illustrated for light propagation modes (see this article) in the following figure, which shows the modes split with growing perturbation intensity $\epsilon$:

