By definition there should be only “one” seed ship, since it’s being used to do a complete restart. So I guess locating it at your homeworld would be most appropriate.
Research, resources and credits are all very low when you start the game, so 10% of your starting empire would be significantly lower than 10% of your empire 6 months later.
I think a tech should be required before a seed ship can be created. Possibly a branch of the colonization tree and should be placed on the tree that would represent the average of a player’s status one month into the game. If you want to restart within the first month, I think it should just be a full reset, you retain nothing, you just start over. It won’t be a specific amount of time (ie, the month), just the placement of the tech on the tree. Someone who’s built a lot of labs in the first month might have it available sooner than the person who hasn’t build a lot of labs yet. (I’m probably not describing this well, I’m sure someone will help me here lol)
The player, who’s been playing 6 months for example, is likely only going to restart because they are expecting to lose their empire. So at the first hint of trouble the player builds their seed ship.
The seed ship is only to provide a transport for a specified number of the population (similar to a colonization ship) and labs, that’s it.
The seed ship is has a flagship rates of let’s just say 16, so you can bring along 15 freighters that could contain more population and resources.
The seed ship can’t be destroyed if targeted for attack, HOWEVER it’s accompanying ships can, so an opponent can’t stop you from resettling, but might choose to make it difficult for your restart. I’d recommend that (and open for debate on the number) the number of labs be limited to 75 with full retention of all previously completed research. This way you don’t lose what’s been completed, but you are limited on the progress on what’s in the queue.
Once the seed ship is built, you would be required to add whatever population you intend to carry (limited by the space allowed for them) and to go collect up to 75 labs. The labs are then removed from the planet they were on and the seed ship is full once it’s collected the 75.
Once the seed ship and fleet is ready to be used for a restart, you’d launch it (by utilizing a specific send function). It would travel x number of light years visible to any opponent in the area who might want to take out accompanying ships (maybe the equivalent of 1 hour’s time), then the entire remaining fleet enters a wormhole (descriptive purposes only here, it’s not subject to attack at this point) and arrives at it’s destination which would be random as you suggested, but a viable resettling area. Now the fleet is again subject to attack with only the seed ship being impervious to destruction and you go exploring for a new home system to settle in.
Once launched, the research that was completed is captured and all planets previously owned are abandoned (just as they would be if you reset). You retain whatever credits you had at launch time. Until you resettle, they are not used, nor have a way to increase as no taxes are collected during the seed ship’s flight time.