Agreed with Ted Bunn's answer, that it is possible to keep a satellite over a fixed time point on earth by using the Lagrange points. One of those L-points is on the opposite side of the sun, which doesn't help much. There are two problems with the other four. One is that there are exactly four solutions, corresponding to exactly 4 local times on Earth: high noon; midnight; ROUGHLY 7am; and ROUGHLY 5pm. The other problem is that all such L-points are far, FAR outside of LEO or geosynchronous orbit.
Another possibility, which would take very little station-keeping fuel, would be to send the satellite into exactly the same orbit around the Sun as Earth. Thus, it would appear to an Earth-bound observer as if the satellite was always above the dividing point of day and night (6:00am or 6:00pm, depending on whether the satellite was leading or lagging earth in the shared orbit around the Sun).
The satellite would have to be much, much farther away than geosynchronous orbit in order to be able to fight the constant pull of Earth. It might be possible to place it far enough out, and close enough inside lunar orbit, to gain enough periodic lunar pull to aid in stationkeeping.
So, six solutions have surfaced, for stationkeeping over a fixed local Earth time with little or no stationkeeping fuel burn: ~7:00 am; 6:00 am; noon; ~5:00 pm; 6:00 pm; and midnight. Happily, one of those solutions coincides with "Margaritaville time."