Giustiniani is a rather obscure figure so it is hard to say definitively what his motives were when he came to Constantinople. His career prior to the siege is hard to pin down but the information we have suggests he came from a venerable Genoese family and served as consul for Genoa in city of Caffa (modern Feodosia) in the southern Crimea. However he seems to have spent much of his life as a solider of fortune engaged in piracy and mercenary work.
As a mercenary his invaluable aid to Constantinople probably wasn't without self interest. One of the primary accounts for siege, that of the Byzantine Michael Critobulus, reports a rumor that Giustiniani had come to the city in response to an offer from the emperor. Supposedly, the Byzantines were willing to give the island of Lemnos to him in exchange for assistance. Lemnos was one of the last pieces of territory that was still under Byzantine control so this was a steep price but one imagines desperate times called for desperate measures.
If this is true then we have a pretty good incentive for Giustiniani. His family seems to have been influential on the island of Chios so adding another Mediterranean island to that sphere of control was probably an attractive prospect. Also, he wasn't the first person to be offered Lemnos by the Byzantines in exchange for help. Alphonso V of Aragon was apparently also offered such an inducement although obviously didn't take up the offer.
We can question whether or not Lemnos was a prize worth fighting for, especially in such a dire situation. In any case, Giustiniani certainly fought hard and was instrumental in the Byzantine war effort lasting as long as it did. Maybe he was fighting for something more than just material advancement but the offer of Lemnos is the best motive I've been able to find.
Sources: The Siege and the Fall of Constantinople in 1453: Historiography, Topography, and Military Studies by Marios Philippides and Walter K. Hanak and The Papacy and the Levant, 1204-1571: The Fifteenth Century by Kenneth M. Setton