It probably wouldn't be called "the Ottoman Empire", since it's named after Osman, which is a Muslim name. But it's possible that if the Byzantines had endured longer, they could have come to depend on Turkish mercenaries. As many Turks were still pagan in te 11th c, these could have adopted Christianity in Byzantine service, and eventually establish a new dynasty. Hence, Byzantine Empire with Turkish Dynasty = Christian Ottoman Empire.
That's actually even more bloody likely than my suggestion, especially since the Byzantines had various dynasties from various different places. A Turkish dynasty would likely have adopted the Byzantine culture, customs, language and faith, and not differentiate itself radically from the Byzantine Empire in general... And depending on the numbers of Turks having arrived in Byzantium, they could have influenced major parts of Anatolia or even the Balkans with their beliefs and military doctrines.
Such a path would have changed the traditional meaning of "Turk = Muslim" for Western Europe too. Probably the term "Turk" would be kept at its Byzantine meaning, meaning nomadic peoples from the East (such as the Seljuks, the Pechenegs and others.) A Turkish dynasty would retain the characteristic name (like "the Macedonians" and "the Constantinians", they could be called "the Turcopoles" or descendants of Turks.)
Changing the thousand-year-old Byzantine system to something entirely new would be major butterflies/ASB however. I imagine that this dynasty would use Greek as the state language and orthodox Orthodoxy as the state faith still, partly due to the strength of the old aristocracy and the like.
Which still adheres to the historical fact that the Ottomans used to integrate existing systems through their conquest, if I am not mistaken...