How does a mortal being make themselves Divine? How did the Nine Divines do it, and how did the Tribunal do it, exactly? and is there any other (theoretical or otherwise) ways?

EDIT: what are the metaphysical processes?

Mantling, which is how Tiber/Zurin/Wulfhart did it to become Talos, along with the CoC with Sheogorath. The Tribunal took Shor's heart and used its power to become gods, thanks to the tools...CHIM, and whatever is in the http://www.imperial-library.info/content/loveletter-fifth-era-true-purpose-tamriel. As for the processes,

As for the Nine, by Alessia taking the nordic and aldmeri pantheon, combining them, then telling everyone to worship them and spread around that [censored] heresy.

There's also another, more difficult option: free yourself from the mundus, and resume your previous godhood.

There's also what ever it is the Hoon Ding uses. It could be argued as mantling (a mortal acting in the characteristics of HoonDing), but it could also be the HoonDing joining the body of the mortal (hence Cyrus is no longer HoonDing once he defeats Septim). It could also be argued that any mortal who is worshiped sufficiently becomes divine via mythopoeia.

Isn't that just a form of mantling?

No...
I don't really see why you think it would be.
Mantling is imitating an extant deity.
What Lady N referred to would be becoming a new deity altogether through collective world-shaping belief.
For example, Ebonarm.

Never played daggerfall, so I never heard of him before. That's where the confusion came from: a lack of examples.

And The Old Ways and An Overview of Gods and Worship both explain that gods are spirits. There are lots of spirits of varying degrees of power and influence, which seem to correlate to the amount of worship these spirits receive from mortals.