PPRuNe Radar, in many cases it is of no time benefit to land short of destination. It can, of course, be very uncomfortable for the passengers (especially the one who's ill) to stuff the nose down to bring it in at, say, PIK instead of LHR when coming on from across the pond with a passenger who's only just been taken severely ill.
You do, however, want to be assured that you'll get priority routing to destination.
Often, there is considerable discussion going on between the cabin crew and the flight crew concerning the passenger's state, as well as between the aircraft and, say, a company doctor, monitoring the passenger, taking advice on what options are open, whether the passenger should be put on the ground at first possible opportunity or whether he can hold on for the extra 20 minutes, (generally the case), balancing all this risk against the extra cost of a diversion, etc. etc.
I have never heard of a crew declaring a spurious "passenger illness" to get in faster. If any ATCO is ever suspicious that they may have been taken for a ride, you can, of course, phone the company's ops department. Any pilot trying that on would get very short shrift indeed in any company for which I've worked (well, except one, and that's another story
).