Should I replace or patch a plaster ceiling?
CAUTION ALERT !  Dealing with a failing plaster and lathe ceiling can be very tricky. Before you make the decision to tear it down, which as Chris mentioned, a messy job, or simply put drywall over the existing plaster, you need to carefully evaluate the soundness of the lathing to the strapping or joists.  What I mean by that is, are the lathes still firmly attached to the joists and the problem is the plaster "keys" failing from around the lathes, or are the lathes themselves falling away from the joists? 

With that said, covering the old failing plaster with drywall is fine. If the lathes are solid to the joists, then putting drywall directly over the old and using screws long enough to reach the strapping or joists will work fine.  However, if the lathing itself is failing, then you must install strapping 16 inches on center, perpendicular to the joists, securely screwed to or ring nailed to the joists. The purpose of strapping is twofold. First reason is to support the failing ceiling securely to the joists. The second reason is that if you just screw drywall to the failing plaster and lathe, the weight of the ceiling will be against the drywall and could pull the screws through the drywall board allowing it to sag or joints to crack now or in the future. 

I must say that even though tearing down the old ceilings is a big job, it certainly gives you a clean start and an open view to correct any old wiring problems, add new lighting circuits, install new flush mounted electrical boxes, and install new level strapping etc.