GCU Grey Area uncovered the following information for us.
The rare Particle Repeater Mk3 is offered in exchange for a "Spacefly Egg Collection", which is crafted from 5 "Spacefly Eggs (authentic)" items. In order to find the latter, think about Spacefly egg lore
Or alternatively, the following spoiler can help.
Spaceflys lay their eggs in nividium asteroids so "Spacefly Eggs (authentic)" items may occasionally be found when they are destroyed by the player or Mining ships.
Note that other items with Spacefly in the name exist so take care to be looking for "Spacefly Eggs (authentic)".
The related mission is called "A Healthy Enthusiasm". After returning to the NPC offering the mission with a crafted collection, you can either exchange it for the opportunity to purchase a Particle Repeater Mk3 or sell it for profitsss.