Flying in a Cessna 182 North of LaBelle VOR, encounter strange light not on radar.Just spent 20 minutes on the phone with Miami ARTCC (Center); would love anyone's insight here. Bear with me, kind of a long, detailed story: Flew back from FXE to TPF direct with my student tonight. All of the Avon Park MOA's and Restricted areas were active/hot, and we could see about 6 F-16's (as per Miami center) dog fighting and practicing intercept procedures at 7000' and above.Upon clearing the MOA and due east of Arcadia, what appeared to be a LED landing light abruptly popped on just above the cloud layer and immediately caught our attention. It appeared to be about 500ft below us, at our 2 o'clock and figured about 5-7 miles, and looked as if it might be in conflict in a few minutes. The light then started flashing, presumably because he saw us up ahead crossing and wanted to make his position known. I turned on our landing and taxi light of the 182 to acknowledge, yet this other aircraft kept turning this light on at odd and random intervals until he went dark for about a minute. Fed up, I asked the Miami controller for a traffic advisory on the aircraft at our 2 o'clock and about 3-4 miles. He responded that he didn't observe any traffic in the area. A little puzzled, I asked him if he had any primary targets in the area and he told me to standby.A long 15 seconds passed by and the light appeared again! This time it appeared to now be at our 1 o'clock but the light still pointed at us (?) and then quickly went out again. The controller came back on and said that he had checked and rechecked his backup systems and assured me that there were NO primary(pure radar return) targets in the area. Just when he said that, an aircraft behind us a few miles asked where we were, the controller told him, and that pilot reported that he was watching all of this happen AS well! Then a third aircraft also chimed in and said he too was seeing the strange light and it was moving across the sky rapidly. Three aircraft in all saw this. The controller was completely bewildered at this point....needless to say, upon landing, I called up Miami Center.I talked on the phone for 20 minutes to the D-Side of the sector I was in, who turned out to be a supervisor and was training the R-Side. He too was sufficiently freaked out and confused along with the rest of the guys on "the floor." He said that in his 27 years, he had odd occurrences like this every few years and they usually turned out to be flares IN the MOA but never in this location, intermittent, and underneath another aircraft moving laterally. He thought that it was a UAV (which is of course illegal to operate in the NAS), and that if it was, it could have been much closer to our aircraft than we had thought (that's always comforting to hear) and it would also explain why it was not on their radar if it was small enough. I chimed in and figured that it started turning its lights on and off in an effort to avoid us.This was not a flare or laser. It changed position laterally and maintained its altitude.