Could the Byzantines have reconquered Cyprus from the Latins? I'm thinking of when they were in exile at Nicea of shortly after the re-occupation of Constantinople in 1261, when they were still comparatively strong.
Hmm. It would have been very difficult to do logistically for starters.
Strategically, the Palaeologan Empire was a state with enemies on all sides: Bulgaria in the north, the Serbs to the northwest, the Normans in the west and the Seljuks to the east. The fact that they chose to ignore the eastern frontier, and allowed the gradual infiltration of Turkish groups into its frontier zone, shows their sense of priorities. For rulers in Constantinople, the Bulgarians and Normans were THE enemies they needed to keep an eye on.
Meanwhile, the distant eastern island of Cyprus, well-run by its Latin rulers, and with compact, efficient, military forces, would have been quite a challenge to attack. If Michael VIII Palaelogus tried when he had a relatively big navy, Western Europe would have been in uproar straight away - and since Michael VIII based so much of his policies on keeping the West sweet, this seems deeply unlikely.
One way I think the Palaeologan Empire could have become involved in Cyprus is through some sort of marriage alliance, which gives the Empire a stake (and supporters) on the island.
The other way is if Michael VIII falls and some historical figure unknown to our timeline (i.e. one you make up!) comes to power after 1261 (possibly someone known to the circle around Patriarch Gregory of Cyprus, who was head of the Orthodox Church from 1283-89 IOTL) and who is maybe a Cypriot and interested in that island. But if he did come to power, or influence in court, and pursued his Cyprus policies, the Empire's enemies in the north and west should have alot of success - unless this character can play them off against each other with truly Byzantine diplomacy...
Are you thinking of starting a thread on this? I'd be happy to contribute if so.