Quite trivial. You start with 5 prestige, you get +1 per each acquired planet and -1 for each lost planet (there are also other sources of prestige). When you reach prestige 0 you automaticly are overthrowned and replaced with "a better emperor". You lose prestige when you have negative treasury and per each turn a battle takes on the imperial capital planet (you are supposed to stop all enemies before they even reach the capital) and for each turn of prolonged rebelion (like if longer than 30 turns -1 prestige per additional turn).

Are there other means to increase prestige?

The way I see this currently, it's a formula that follows this concept:

Currently Controlled Planets + 5 - X turns with negative treasury - X battles in orbit of capital - X turns of rebellion

Essentially, aside from planet count, you have 5 lives to spend on negative treasury, capital battles, and rebellions.

Is the intent to force players to expand their empire to survive loss of prestige?

Yeah, sure, probably. But it's not very relevant. You are to crush the rebels (organization), not their supporters (part of the population).

I was referring to mechanics only. If population counts towards anything (taxes?) the sheer amount of population that pays taxes would be ultimately reduced when you quell a rebellion. The point can be moot, as you pointed out, because rebellious planets don't pay taxes, so you're just cutting your losses by rectifying this situation, correct?

Are there other means to increase prestige?
The way I see this currently, it's a formula that follows this concept:

Currently Controlled Planets + 5 - X turns with negative treasury - X battles in orbit of capital - X turns of rebellion
Essentially, aside from planet count, you have 5 lives to spend on negative treasury, capital battles, and rebellions.
Is the intent to force players to expand their empire to survive loss of prestige?

First of all, an Empire needs prestige, it's kind of "compatible with the overall mood" There is an Emperor, there is an Imperial Throne and there is a Prestige.

I didn't plan prestige as a tool to achieve some design goal, it's just the primary metrics (score). But of course, I could use some side effects of the prestige to assure some non lethal punishments of the player (like negative treasury).

Overall, I want to reward the player for doing well (expanding, making audiences and generally behaving like a decent Emperor) and punish for being incompetent (like ignoring rebellion or degenerated finances). I haven't thought of other sources of Prestige, but I think there will be some.

I was referring to mechanics only. If population counts towards anything (taxes?) the sheer amount of population that pays taxes would be ultimately reduced when you quell a rebellion. The point can be moot, as you pointed out, because rebellious planets don't pay taxes, so you're just cutting your losses by rectifying this situation, correct?

Rebellion is bad. If you don't crush rebelion in time you will be disposed (maybe not directly, could be by use of prestige system or avalanche effect). Population is not the key issue here.

Note: it's just my vision, and have not coded rebelion yet, so it all can change, post yours.