Here's a straightforward design, intended to be built in a 100 k-ton slip:
Scavenger class Asteroid Miner 96,050 tons 1858 Crew 2499.2 BP TCS 1921 TH 0 EM 0
1 km/s Armour 1-186 Shields 0-0 Sensors 1/1/0/0 Damage Control Rating 1 PPV 0
MSP 16 Max Repair 120 MSP
Cargo 5000
Asteroid Miner: 18 module(s) producing 360 tons per mineral per annum
Fuel Capacity 5,000 Litres Range N/A
This design is classed as a Commercial Vessel for maintenance purposes
Duranium 1272.9. Corundium 1080, Mercassium 79, Neutronium 62.4, Corbomite 5. Your milage may vary with Armor tech.
Assuming a 20 mining-skill ship's captain, a 10 mining-skill Asteroid Administrator, and a 20 mining-skill Sector Governor (who passes 1/4 of that skill down to subordinates), this ship will recover all the minerals spent in its construction in ten years... if it is working an Asteroid with ONE mineral on it at 1.0 concentration. Naturally, if it is working a five-mineral Asteroid, it will recover the investment in two years.
That's at base (10 minerals per mine) tech level. Naturally, your tech will probably be better than that. The displayed "360 minerals per annum" is at my current mining rate of 20. Treat it as 180 per annum for base tech level.
With no ship's captain, no planetary governor, no sector governor, no tech boost... it takes about 50% longer to recover your investment.
So 15 years for an absolutely base-level version of this ship, which gains no bonus from any source (leaders or tech) and works a rock with only one mineral... reduced to a single year if it gains a decent bonus from tech, captain, planetary and sector governor and works a fairly rich (five-mineral) deposit.
NOTE: An Automated Mine costs 240 minerals, and under the same assumptions (no tech, no governor, single mineral, etc) requires 24 years to pay that back.