A particularly ambitious or vengeful Archon begins to amass followers. That's basically it.
Most "new" Kabals either come from Low Commorrites banding together so they don't get pushed around by the bigger Kabals, or a High Commorragh Kabal gets destroyed and the Archons and Dracons within it splinter off and start up their own Kabals.
Your idea is pretty common too, one of the biggest dangers for an Archon is their Trueborn consuls getting too high and mighty and peeling followers from the Kabal as a whole to them. A revolution to put the aspiring Dracon into power, your armory being raided as half the Kabal flees to a different part of the city, they sell secrets to a rival Kabal which puts you under attack, etc.
Basically, the beauty of the True Eldar is that any way you can imagine someone rising to power is probably totally viable. Politics and backstabbing make Game of Thrones look like a sitcom, your imagination is free to run wild.
Unless it involves Vect. You don't frak with Vect or the Black Heart.