I want to hang something like http://www.skychairs.com/ or similar in my living room. If the joists were exposed I would mount a beam between two of them and hang over or through that. However, when the joists are hidden behind drywall or sheetrock or other ceiling material this doesn't seem to be an option. Putting an eye screw into the middle of a joist might be ideal, but scares me in a few ways (strength of the screw, weakening the joist, unscrewing over time, etc). What is the best way to approach this problem?
Just because you can't see the joists doesn't mean you cannot use your beam approach for exposed rafters. The structure is essentially the same, except for a thin layer of sheetrock between the two, which has little impact on the forces involved. The best thing to do is distribute any concentrated load over multiple connections. Use a flat 2x4, or better yet 2x6 that's long enough to span 4 joists, about 4 to 6 feet. Place the chair connection at the center using a large through eye bolt, using an over sized washer to spread the load into the wood. The nut and bolt end will end up punching into the ceiling, but that's OK. Holes are easy to patch if the beam is ever removed, otherwise, it doesn't show anyway.
Attach the 2x beam to each joist with two 1/4" dia. lag screws. Use a screw long enough to penetrate 2 inches into the joist material, probably 4 inches. Drill a 5/32 inch pilot hole before placing each lag to prevent splitting the joist. The resulting 8 screws is more than adequate to support any adult and will not significantly weaken the joists.
Drywall is a repairable material. When needed, be prepared to rip it open, do what you need to do, and patch it afterwards. If you are not absolutely sure how the ceiling is built, you can be lead astray following lines of nails/screws in drywall, if the ceiling was strapped with 1 x 3 running crossways to the joists before drywalling, rather than direct-attached to the joists. A studfinder or a magnet locating the drywall nails/screws will show the 1 X 3, not the joists.