Medusa Miners increase the production of all buildings in the "Extraction" section of a planet. The multiplier applied to the output of these buildings is...
Multiplier = 1 + Nmm * 0.2 * LN(2) / LN(2+Nmm)
...where Nmm is the number of Medusa Miners at that planet.
This multiplier only applies to materials that are actually extracted and do not apply to those same materials that are merely manufactured. For example, if 100 Medusa Miners (Multiplier is 3.9974; remember that a 299.74% bonus is a 3.9974 multiplier) are in orbit around Tataridu, then the output of the Oil Pumps are multiplied by 3.9974, but the output of the Biofuel Refineries are unaltered.
Medusa Miners have a speed of 0.5 and no storage capacity. To increase the traveling speed of Medusa Miners, it is necessary to merge them into a fleet of ships that has storage capacity and then have the merged fleet carry Engines.
Construction of Medusa Miners requires Space Mining researched to Level 1 and a planet with Shipyard 18. Each Medusa Miner requires the following materials (before any deductions from Karan Art of War):
Iron: 300 Million
Steel: 1 Billion
Titanium: 100 Million
Nanotubes: 1 Million
Robots: 100 Thousand
Full Battery: 250 Thousand
Each Medusa Miner has 2 hitpoints (before Quris Art of War) and a Weight of 100 Million. Be certain that any Medusa Miners have been split out from a combat fleet prior to attacking any enemy fleets.