This should be a fairly simple way to give asteroids and dead planets an purpose in game.
Proposal: Every asteroid and dead planet in range of a economic starbase boosts raw production of the colonies in that starbase's area of effect by an additional +2%, over and above the base of 10%.
Other Starbases: Other types of starbase grant a boost to military production to whichever Shipyard it is fed too, at a base of 1% per dead world and asteroid in the are of effect. This boost would decrease over distance the same as current shipyard falloff, and a max of 5 can feed a single shipyard. Makes the shipyard falloff Pragmatic Ideology trait more useful, not to mention the relevant trait.
Suggested: To reduce constructor and starbase spam, limit any planet to only benefit from the best starbase in range in any category - raw production, research, industry, morale, etc. Also, each asteroid / dead planet would feed a single starbase, with Economic > all others, as raw production > military production.
Suggested: Which starbase gets a resource (incl strategic and relic) should default to a) whoever had the most influence and 
whichever has the most modules of the correct type.
I've had plenty of starbases with 5 planets + asteroids in range, so that is potentially a +25% boost to military production of a shipyard, or a boost of raw production from economic starbases from 10 to 20%.
You don't need to add a new type of module, unless you want too (too close to release). If you did, asteroid mining modules would not require a base to be specialized, they would be in addition to the base type, like archeology modules, and should boost 1%/2%/3%. It would probably be better this close to release for this to be inherent in starbases, and auto feed the closest shipyard, and add a screen to the shipyards that lists the closest non economic starbases.