Basically this is a theory that doesn't have any evidence in the game and is just me taking advantage of plot holes and things left unsaid. (feel free to point out where I went wrong. I haven't looked at timelines in depth to see if my theory is plausible)
So basically the background is this. Three street urchins (an untrained mage (Ria Silmane), a Street thug (Jauffre) and a pickpocket/thief (Caius Cosades, younger than the other two)) attempt to rob a man, (Uriel Septim in disguise, off to sleep with some wench possibly Martins mother) only to be foiled by his bodyguards. Uriel takes pity on them and decides to adopt them into his household (judging from his impulsive nature pre-Arena this fits imo). Jauffre and Caius join the Blades and Ria is apprenticed to Jagar Tharn.
After a few months/years Jagar Tharn kills Ria and imprisons Jauffre and Caius in the ancient dungeons, and Ria's ghost helps them escape to High Rock through the Shift Gate, If I end up writing this as a fan fic Caius will be aroun 15-16 and won't be all that helpful earlier in the quest. The ten year span of Arena happens and Jauffre and Caius defeat Tharn and rescue Uriel who names Jauffre the Eternal Champion and grants them both the Septim name (maybe a bit over the top in bending the canon?).
General Warhaft steps down and Jauffre becomes head of the Blades with Caius as his second in command/spymaster. (I retconned Caius to be a thief, not a monk, as it makes more sense). Caius screws up big time by either letting one of Uriels sons die on his watch and/or degrading relations with one of the provinces (probably Summerset). As punishment he is stripped of the Septim name and exiled to the Illiac Bay area where he performs the feats of Daggerfall. Upon completion of this task he is exiled to Vvardenfell, albeit as a high ranking member of the Blades, and Jauffre retires as Head of the Blades and retires to Weynon Priory.
The Nerevarine arrives in Morrowind and Caius guides him. When riots in the Imperial City start (as referenced to in dialogue) Uriel recalls him to make amends and enlist his help. Caius dies protecting the Emporer or returns to Morrowind after the Dagoth Ur is defeated.
So, that's about it. Feel free to suggest more canon friendly options (I know it is stretching it thin as it is), or pick it apart.