Asteroids represent an immense treasure trove of both scientific knowledge and mineral resources waiting to be fully exploited by human explorers. However, reaching even a near one with just a robotic probe would take months of travel time and cost hundreds of millions of dollars. A manned mission would push current space technology to its limits and may prove cost prohibitive.
However, another option occasionally mentioned is not to send probes and astronauts out to an asteroid, but to bring an asteroid into Earth or Lunar orbit for much easier access. The idea has been seen in various science fiction sources, such as the Gundam anime series, the Night’s Dawn Trilogy by Peter F. Hamilton, and the novel Eon by Greg Bear. To the best of my knowledge, however, no one has ever done a serious scientific study on this possibility.
Comets instead of asteroids could also be diverted into the Earth-Moon system. However, because of issues with outgassing so close to the Sun (the dust and ionized particles given off by the "tail" would be captured by Earth’s gravity and could compromise radio communication and satellite viability), they are usually not considered to be practical for this scheme.
Any number of the methods mentioned in the related article Redirecting Asteroids (link below) can be used to alter the trajectory of an asteroid in order to allow it to eventually be inserted into Earth orbit. A suitable asteroid would first have to be found; one or more probes would have to be sent out to very carefully study the target to determine its exact dimensions, composition, and density. Once this is known, mission planners will have a much better idea of where to place the initial and subsequent propulsive impulses that will move the body without fracturing it.
Candidate asteroids would ideally be small, at most only a few hundred meters across, probably considerably less. They would also need to be located relatively close by. Earth has a number of ‘quasi-satellite’ asteroids that may be suitable, small rocky bodies that have orbits around the sun that closely mimic Earth’s but are not in Earth’s orbital plane. Earth’s gravitational influence occasionally tugs one of these enough so that they enter into a semi-stable orbit about the planet for several decades or centuries at a time. Other possibilities may be asteroids that sweep in close to Earth’s position, such as asteroid 99942 Apophis will do in 2029, that can be diverted into position with just the right nudge in its solar orbit.
Where exactly to place the diverted asteroid in the Earth-Moon system is debatable. Low Earth orbit would likely be too much a risk for even a small space rock, and the political fall-out of having a potential strategic weapon orbiting closely over everyone’s heads would be considerable. Also, maintaining so low an orbit would need constant station-keeping, something that could prove problematic and expensive for a million or more tons of mass.
Longer-period orbits, such as Geosynchronous Orbit, are a more attractive option. The diverted asteroid would be far enough away to ease the worse of peoples’ fears, but would still be close enough to allow fairly easy access from Earth. However, it may still be too close for comfort for many people concerned about a mishap or a hostile action sending the asteroid hurtling Earthward. Also, station keeping would still be a necessity (though less frequently needed than in LEO), and constantly fine-tuning the position of an enormous rock could still prove a daunting matter for the long term.
A more equitable solution may be putting the asteroid at one of the Earth-Moon Lagrange Points. L1 may be a desirable location, as any asteroid there could serve as a staging base and resource for future lunar exploration. Unfortunately, L1 is only metastable, and orbital adjustments would have to be made on the order of at least once every two weeks to keep the rock in place. Again, with a mass of a million tons or more, this may prove prohibitively expensive.
Lagrange points L4 and L5 would appear to be perhaps the best location in the Earth-Moon system for a redirected asteroid. Both points offer true stability, and little station keeping would be necessary once it was in place. Also, at a quarter million miles distant, the asteroid could seem far enough away that most peoples’ fear of an accidental impact with Earth would greatly diminish. The down side is that the asteroid would still be far enough away to require at least several days’ journey, and boosting needed equipment, habitats, and such up to it will be more difficult and expensive than with lower orbits.
A final possibility is to park the asteroid in lunar orbit. This way, if its orbit for some reason did decay, it would spiral in and impact the Moon instead of Earth. Again, this could go a long way toward easing anxieties back on the homeworld of having a potential weapon of mass destruction hanging over their heads in the nighttime sky.
No matter where the asteroid is redirected to, it could serve a number of useful purposes. Most obvious and immediate would be scientific. So close, even amateur astronomers could catalog and investigate almost every facet of the object, allowing us to learn a great deal more about the nature and composition of asteroids in general. Manned missions to it would be inevitable as well, and an asteroid could easily become the second celestial object that humankind ever sets foot on.
A number of the grander visions for orbital structures, such as solar power satellites and O’Neill colonies, would require huge amounts of material resources to construct, resources that would be very cost prohibitive to launch directly from Earth or even from the Moon. Having a large source of raw materials already in orbit and in microgravity could greatly alleviate such costs. However, only asteroids with heavy metallic content, such as the nickel-iron variety, would probably be worth the cost of exploiting this way.
The asteroid could of course also have a manned base built on it, or more likely, in it. Many meters of natural rock would serve very well as a shield against solar and cosmic radiation, and could enable long-term stays beyond Earth’s protective magnetic field without fear of potentially health-damaging exposure. Farther into the future, larger asteroids may be fully hollowed out, their interiors terraformed, and spun for gravity, creating Hollowed Asteroid O’Neill colonies. In fact, because of this, asteroids could offer a better and more viable option for creating deep-space outposts and colonies than building such up from scratch.
If its to be used mainly for habitation, a carbonaceous ("stony") asteroid may be preferable to a nickel-iron body, as the former should prove easier to drill into and excavate.
Asteroids diverted into the Earth-Moon system could serve other purposes. A small asteroid may be used as a counter-weight or a geosynchronous station for a Space Elevator system. A body with a significant amount of water or methane ice could be used to provide a moon base with much needed volatiles or water. On a more sinister note, an aggressive power could divert asteroids meant as impactor weapons to "holding orbits" in order to threaten rivals on Earth with a possible full-scale barrage.
No matter their intended purpose, diverted asteroids in orbit would easily be visible in the night sky to the average person. They could have significant psychological impact on the world population, who would have to get used to the presence of this new moon.