Very interesting Fraese.
Would these quests be secret?
If so, have you ever played Risk with secret missions? Suddenly someone says: I control all of Europe and thus I have won the game because it says so on my special mission card.

o Well that is of course more serious than what you suggest since it decides the whole outcome of the game, who the winner is.
If all missions where public I would like this idea better.
If they were public you would be able to guess what your opponent was trying to do and could try to outsmart him. But when his quest is unknown it becomes a lottery.
I will try and invent a few quests that could be public.
1. Their are inbuilt VP-rewards for certain discoveries, such as Cape Horn and Australia (you even get a message when you do it so there can be no doubt about who earned the distinction of being the first). That could be rewarded with prestige.
2. The one who first reaches a certain number of owned provinces (e.g. 20, 50, 100)
3. The one who has zero bad boy points (that would get prestige in RL I believe) at session's end. Or perhaps simply: the one who have the fewest BB points at session's end.
4. The one with the most +100 relations with other nations... or other electors...
6. The one with the highest number of manus (a reward in itself of course)
It could be a good idea to look in the rules section for Thirst for Glory II. The concept of prestige was vital for that game and its predecessor.
Elio had a list of rewards for different things, like the one who first reaches infra 5 (as if that was not a reward in itself).