I don’t think it’s necessarily anything, really. I don’t think any of the stats are anything. Think about speed, for instance. Rapidash has a base speed of 105, and that’s clearly because she runs really fast, but then, Kadabra also has a base speed of 105 and it’s equally clear that this does not mean Kadabra can run as fast as Rapidash. In Kadabra’s case it represents something more like reaction time. So really speed is “anything that enables a Pokémon to gain a tactical advantage which can be represented by moving first.”
Special defence is “anything that can mitigate the HP loss caused by a successful special attack” (and of course “HP” could represent a bunch of things like the will to keep fighting, or the ability to turn a direct hit into a glancing blow, as well as actual physical endurance). Amnesia and Calm Mind increase special defence, while it can be reduced by Psychic, Fake Tears, and the awful grating noise of Metal Sound, so on that basis it seems like it can represent something like focus or willpower. On the other hand, Shell Smash, Close Combat and Acid Spray can reduce special defence, and Defend Order increases it along with defence, so it seems like there is also an element of ‘physical’ resistance to energy-based attacks in there. Most of the other stuff that interacts with it is just ‘magic’ of one kind or another – Shadow Ball, Geomancy, Cosmic Power and so on.
Some Pokémon whose highest stat is special defence include Regice and Cryogonal (transparent prismatic body reflects and diffracts energy attacks?) Lugia and Ho-oh (’sacred’ status blocks hostile magic?), Probopass (magnetic fields disrupt energy attacks?), Goodra (gooey body absorbs shocks?), Mantine (veil of sea spray dampens energy attacks?), Umbreon (extremely great willpower?), Sylveon (defensive magic?), Milotic and Florges (aura of peace reduces opponents’ will to cause harm?), Tentacruel (gems reflect attacks?), Mr. Mime and Hypno (mime and hypnosis abilities grant incredible focus?). I don’t think there’s any one thing you can point to and say “high special defence means a Pokémon is good at this.”