If he surrenders, his legacy is immediately shot to hell compared to OTL. As for the remainder of his life, he'll move to Morea, where he will be an emperor in name only living at the suffrage of his younger brothers, both of whom have more land under their direct control than he does now, and both of whom themselves live at the suffrage of the Ottoman Empire. Worst case scenario he is immediately killed by his brother Demetrios, who assumes the Imperial crown and plunges the Morea into civil war, as his brother Thomas will certainly not accept this. Demetrios calls on the Ottomans for aid, as he was pro-Ottoman, and the Ottomans use this excuse to take control of the Morea as IOTL.
If we want to get a little more interesting, we can have Demetrios killed off before he can kill Constantine, either naturally, by assassination, or in a well conducted civil war, as both Thomas and Constantine were far more capable military commanders than Demetrios (OTL he was beaten by Constantine and Thomas on different occasions, and was well known for his incompetence). Realistically there is no future for an independant Morea with Demetrios alive. He pretty much spent his whole life weakening the last feeble efforts of the empire, and was Angelos level incompetent. If he survives his only purpose will be to serve as an excuse for an Ottoman invasion, and given a convenient excuse they will jump at the chance. Without a convenient excuse, there is at least a ghost of a chance of Mehmed leaving them independant for the remainder of his reign (unfortunately, just how much of a chance there is can't be known with certainty, as both the Morea and Trebizond gave him convenient pretexts for invasion within a decade of the fall of Constantinople IOTL).
Now, we have Demetrios dead and burried and can begin to focus on the Morea. Thomas and Constantine were generally on good terms with eachother, and had each spent much of their careers as Despots of Morea, and were also both Pro-Europe, so personell is good and closer ties with the merchant republics is almost a given.
From there it's all in the hands of other nations. Throughout Mehmed II's reign there were numerous close calls and potential coalitions that could have effectively opposed him, but a combination of solid leadership and good luck saw Mehmed through the more turbulent part of his reign until he was the undisputed master of the balkans and Anatolia. The first bullet he dodged was the person of his Grand Vizier Candarli Halil Pasha, whom I made extensive use of in my own TL. Candarli was a conservative who fiercely opposed further expansionism and personally hated Mehmed. He was representative of the old Anatolian aristocracy at the time, and had a considerable support base, but Mehmed, in perhaps one of the greatest political moves of his career, used the siege of Constantiople as a smokescreen, under which he had Candarli assassinated immediately after his victory. Here, he won't have any such opportunity, and thus Candarli may stand a chance of living a good deal longer, perhaps surviving as a foil that keeps Mehmed's ambitions in check, or perhaps things will degenerate and, in the absence of a convenient time and place to assassinate him, Mehmed is forced to wage civil war against Candarli. Either way things are worse for Mehmed than IOTL, and thus better for our Byzantine enclave.
The second chance to nerf Mehmed is at the Battle of Belgrade in 1456. Janos Hunyadi won a great victory over Mehmed with a much smaller force when the latter attempted to storm Belgrade, with Mehmed himself being wounded and his forces threatening to mutiny against him, but following the victory divisions in Janos' forces prevented him from immediately going on the offensive, and Hunyadi, arguably the most serious foreign threat Mehmed ever faced, died of a plague within a month. If Mehmed dies of his wound, then the Ottoman Empire most likely goes into civil war, as his heir Bayezid is only a child, and this could present opportunities for the states on the fringes of the Ottoman Empire, including our Byzantine enclave, which could potentially take control of the Duchy of Athens, at that time an Ottoman vassal, and a previous target of expansion by Constantine XI during his tenure as despot of Morea. Further Byzantine expansion is unlikely, but further Hungarian expansion, as well as potential Wallachian expansion under Vlad the Impaler (freshly installed on the throne of Wallachia) and Venetian expansion into strategic costal areas and propping up the Greeks and Albanians.
In the east, the powerful Karaman Beylik and the Whiite Sheep Turkmen also have a reasonable chance of making gains at the Ottoman Empire's expense. Genoa is almost irrelevant to the greater fate of the region by this point, as is Naples, as both of them are entwined in the political web that will become the Italian wars in a decade, so they're out, though crusading forces are a viable possiblity if the pope is energetic enough (hopefully not Callixtus III).
Long term, there is always a slight chance that, if the Ottomans lose Constantinople to another power, it could be turned over to the current Byzantine heir, but only if the city will be difficult for its conquerors to hold. Further Morean expansion is unlikely except in the event of a total Ottoman collapse, and even then Venice will probably be snatching up the best parts of the Ottoman Empire (Thessalonica, Smyrna, possibly Constantinople). Against Venice, the Byzantines would have a solid chance on the ground, provied there aren't any rival contenders to the throne for Venice to fund, although at sea they're toast. The longer the Byzantines last in this scenario, the better, as a rise of nationalism with a still extant empire would do wonders for them, as would a stronger Russia, a backwards Ottoman Empire, a united anti-Islamic Spain, a resurgent Persia (especially if it's still Shiite), and increasingly powerless Italian merchant republics.
Still, 95/100 times Constantine XI doesn't manage to take Demetrios out cleanly, and it all ends the moment he calls for Ottoman aid.