What is "normal" about normal frequencies and normal modes in coupled oscillations? So, my question is what does the "normal" part mean when one talks about normal frequencies and normal modes in coupled oscillations.
Does it have to do with the normal coordinates that one uses when solving some problems, or with normal in the sense of orthogonal?
 A: "Normal" in the context of oscillators simply means "periodic" – periodic solutions and the frequencies and other aspects associated with them. It's like in "he breathes normally" – the breathing seems to be periodic.
"Quasinormal ones" are those whose time dependence is $\exp(-\Gamma t) \sin (\omega t)$, i.e. they have some exponential decrease aside from the periodic function. The exponential decrease is the "damping".
There is a relationship between the "normal" as "periodic" and "normal" as "orthogonal": the normal (periodic) modes are normalizable to the delta-function, they may be used to construct a (continuous) orthonormal basis.
A: I think "normal" means also "proper to the system", i.e., existing after the system ceased to experience an external force.
A: "Normal" in this case refers to their orthogonality, although the terminology is somewhat ambiguous.  For example, you may sometimes see modes, or more generally basis vectors of some abstract function space, labelled as "orthonormal".  This indicates that they are both orthogonal and "normal" in the sense that they are normalized such that the inner product $ <\psi_n,\psi_n> = 1$
