I thought it was kinda like Go Fish... except now I've looked it up on Wikipedia it's buggerall like it.

One country is given the prize. Don't know when... maybe 2 days after the start of the set.

2 alternatives...

1) The aim is to keep the prize to the end of the set.
You lose the prize either:
by not playing turns for more than 24 hours - it would be randomly given to a player that played within the past 6 hours.
75% chance of losing by being landgrabbed by an SS
by a new spy op to "Search and Steal the Prize"

No spy op can just "search for the prize". It has to be search and steal.

Mil strategy tech will improve chance of SS taking the prize.
Medical tech will reduce the chance of SS taking the prize.

If someone takes the prize, you cannot attack or spy them for the prize for 36 hours. (normal spy ops possible)

If the prize changes hands, a notification is set to all countries to say that "The prize recently changed hands". This notification would be sent up to 2 hours after the prize changes hands.

2) The aim is to not have the prize at the end of the set and would have roughly opposite rules to 1).

Interesting idea. I think it'd only work in a shorter reset length period with a lot of turns though. Maybe like 1 week set with 120 turns a day or so.

I was thinking that it could just be an addition to primary as a kind of side game.

Maybe there could be a bonus to having the prize which would kick in only after at least 1 successful defends, eg enhanced luck or increased explore rate. It could have an increasing effect per successful defence. For example, at
0 successful defences, bonus = 0%,
1 successful defence, bonus = 1%
etc

The reason I thought of it was in order to encourage grabbing. If people were informed that the prize had passed hands within the past 2 hours they could search for attacks in the news/eestats to discover a list of potential prize holders and then go after the prize themselves.