(More crazy space schemes 8-)
Mare Tranquility, as in 'the sea of', where Buzz Aldrin took one small step, is smack dab in the middle of the launch zone for L-2, the outer Lagrange point, way away from Earth, and a good spot for building stuff, possibly. But I'd like to see a solar farm built up in the sea, for symbolic purposes, and a lunar space elevator could be dropped down from L-1 to the near-point, with a cone of 'side-spurs' to any point on the Nearside at all, the better to establish a 24-7 power production capacity. Two locations on the eastern and western limbs of the Moon really should do the trick, with interrupted production about every two weeks as the two areas switch over, one to the other. The solar farms could spread away into the lunar farside, with East and West poles growing wider over time. Receiver/transmitter stations at the remaining Lagrange points, spread out 120 degrees apart in the Moons' orbit, would cover the entire globe up to the higher latitudes, over three-quarters of the planetary surface and most of the industrialized world.
What would it take to put cities and industry on the Moon? Robots, really. Some people to troubleshoot, but construction, production and maintenance could mostly be done from Earth, or at least safely underground. Melting and doping the regolith for cheap photovoltaic surfaces is certainly something that could be automated, like a roomba, or a farmer plowing, sowing, spraying and harvesting his fields. Melting the surface layer, perhaps one cm deep, turning regolith into melt, would mean raising the temperature from an average of 270 Kelvin, -3 C, say to 10,000 F or 5,556 C.
I don't know how much energy it takes to melt rock, but if we were raising 10,000 pounds of water by one degree (or one ton by 5 degrees), that'd be 10,000 BTUs, or 34,100 watts. If each cc of regolith is ~3 mg, and one square meter is 30 grams, then one hectare, or 10^4 m^2, is 300 kilograms, 663 pounds of rock. Rock is not water, but this is all guesstimation, so bear with me; we're talking about 23 megawatt/s to melt one hectare of regolith. Call it 36 megawatt/s or ten kilowatt/hrs, per hectare, 10^3 KW/hr per square kilometer. Even at 10 W/m^2, each km^2 generates 10 MW; 50 W/m^2 and 50MW/km^2 is only ~4% efficiency, enough to melt 50 km^2 per hour, over 17,000 km^2 per lunar day... I'm doing something wrong! Assuming that we need to melt 10 cm of rock and have huge heat losses of two orders of magnitude, that's still 17 times, so that an operation could double itself 17 times during the half month of light. If ~80% of production goes towards generating revenue and two equal shares of energy are spent elsewhere in the operation, the doubling still happens once a month, four thousand-fold a year!
If the receiver/transmitter antennas are part of the works and a phased array, with some upkeep, but the area of production grows only at the edges, and with the manufacture of a fleet of dead-simple, dirt-cheap robots, it might only grow 1 km per month... that'd be 12 km in diameter in one year, about 22 GW, 2.2 TW in ten years time! After the first year, we're talking, @ $.09/KW/hr, $2M per hour, $17.5B a year, $70B after the second year, $160B after the third... ~$500B by the end of the 5th year, when production is over 1/2 TW and climbing. But there's really no reason why the thing can't spread out along the equator at 100m/hr, ~35 km per month, 840 by 24 km the first year, or ~5,000 km^2, 250 GW, $23M worth of electricity per hour!
How much power do we need to run the world? 100 KW/person? 150? Call it one million billion watts, or 20 million km^2, half the surface of the Moon. After 7 years, the East and West Patches touch and are 168 km wide, 1.8 m. Km^2 (over 4% of the surface), 90 TW or ~$8B/hr, ~$70T per year... The patches would probably be considered a blemish, visible to anyone looking up at the Moon, widest and most noticeable at edges, and producing 13 KW for each of those teeming billions complaining about it! Even if it was limited in growth on the Nearside, that's still enough power to completely change industries and lifestyles.