Follow-up question: why did the Eastern Romans keep tabs on where Ayyub, Muhammad's standard bearer, was buried well enough for the Ottomans to find once they took the city of Constantinople?
Not a historian but I study Islamic theology. I will try to answer this question, since this issue specifically is of importance in the Islamic traditions.
During the time of Prophet Muhammad in the early seventh century, he made prophecies that the Muslims are going to eventually conquer many places. For example, during his Hijra (the immigration from Mecca to Medina) he promised one of the men that they are going to conquer Persepolis during his life time, I translated the story from Arabic in Reddit before, but Islamic scholar Anwar Al-Awlaki did a better job translating the story here.
" ... Suraqa then declared that Muhammad’s religion would prevail one day and requested Muhammad to hand him a written promise that he would be honoured whenever Muhammad became the head of the Islamic state. Abu Bakr wrote the promise at the behest of Muhammad on a bone and handed it over to Suraqa. Even as he began to return, Muhammad told him that he would one day wear the bangles of Khusrow of Persia. He asked in wonderment if Muhammad meant the bangles of Khusrow bin Hormuz, the Persian king. Muhammad nodded in acknowledgement .... Years passed. At one time Suraqa became ill and was dying. His house prepared for his death (643 CE/22 AH). All of a sudden Suraqa open his eyes and inquired what his family was doing. They told him that they are preparing for his burial because they dont think he will survive. He ordered them not to worry about him because he will not die. Family member were surprised and asked how he is so sure. He said that Prophet Muhammad assured him of Khusrow's bangles and had not get yet. Persia was conquered during the caliphate of Umar ibn al-Khattab. Umar received a missive from Saad bin Abi Waqqas, commander of the Islamic forces about the conquest of Persia. A fifth of the booty was sent to Umar in Madinah. It contained the famed crown of Khusrow studded with rubies, a gold belt studded with pearls and the gold bangles. Umar was looking at each of the items sent by the Commander of the forces. Suddenly Umar called Suraqa to him and put the crown on his head, helped him don the gold embroidered dress and gold bangles.
Suraqa felt uncomfortable in the royal outfit. He raised his hands towards the sky and prayed: ‘O my Lord, how could I put on something which your Prophet deserved far more than I? I smell the air of punishment in it."
He then turned towards Umar and asked him to distribute all those royal appendages among the Muslims."
(It is of note that the authenticity of this story is not agreed upon by Islamic scholars of Seera (the life of Prophet Muhammad), however I am mentioning it here to show you the importance of such prophecies to Muslims. This story is mentioned in several Arabic sources, most important of them Kitab Al-Umm by Al-Shafi'i (4/157))
Now to answer your question, why was Rome specifically so important for Mehmed II?
Prophet Muhammad made another two more important prophecies, the first one:
"You will conquer Constantinople. What a great leader he will be and what a great army it will be."
(This was narrated in the The Musnad of Ahmed ibn Hanbal, and is considered by Muslim scholars to be much more authentic than the previous story.)
Stemming from this prophecy and the praise of Prophet Muhammad to the army and leader that will conquer Constantinople, many Muslim leaders and armies tried to conquer this city, all wishing to be those mentioned in the prophecy. To this day all Muslims carry great admiration and respect to Mehmed II because of this prophecy and this conquest was one of the main reasons why the Ottoman rulers had legitimacy in the Muslim world as a whole and why they could claim the title of Caliph (Muhammad's Successor, the Supreme Leaser of all Muslim People) with little resistance, although they do not come from Arab origins.
The second important prophecy was the conquest of Rome:
Abdullah ibn Amr said: We were sitting around Prophet Muhammad writing, when he was asked: Which city is going to be conquered first? Constantinople or Rome? Prophet Muhammad said "The city of Heraclius will be conquered first"
(There are many other prophecies mentioning the conquest of Rome but this one is the most authentic, narrated by Al-Dhababi in his book Al-Talkhees.)
Although there was no specific praise by Prophet Muhammad to the army that will conquer Rome or its leader, Mehmed II and many other Muslim leaders were seeking the honor of fulfilling the Prophecy by conquering Rome. Although, in the case of Rome, many of the prophecies are foretelling that the Conquest of Rome will be preceded by a war "in which 99 of each 100 will die, that the victorious army will not know what to do, whether to celebrate victory or grief those who died". Some Muslims believe that this is an omen of a nuclear war. This war is also believed to be followed by the coming of Al-Dajjal, the False Messiah and the second coming of Jesus Christ, the True Messiah.
Note: This is my first answer in r/askhistorians , I am sorry if my answer was lacking. Many of the sources important in answering this question are in Arabic, and I tried to translate them to the best of my abilities. If anyone wants the full Arabic text of the quotes I mentioned, I am very willing to provide.
Mehmet cared about imperial tradition a great deal, in fact, he had the Byzantine Orthodox patriarch crown him as the Caesar of Rome (Kayser-i Rum) although the title seems to have been later abandoned (but only hundreds of years later). Mehmet Fatih sent Ottoman troops marching from their Italian posessions in Otranto in 1480 upwards towards Rome, while troops chanted "Roma'ya" ("To Rome") rather than the common "Allahu ekber" - this might also be related to the fact that many of the troops were from the territory of today's Albania, and were either recently converted or still Christian themselves.
In architecture, they attempted to harness or surpass Byzantine art, and after the conquest of Constantinople we see a tremendous transformation of Ottoman architecture away from things like the Bursa Grand Mosque, with its Seljuq style multiple domes, to the single-domed works of Mimar Sinan, like the Suleymaniye and Selimiye mosques, which imitated the Agia Sophia. It can be argue that a policy of reestablishing Roman heritage remained a part of policy through Selim I's conquest of the Arab world, and Suleyman the Magnificient's plans to conquer Rome. They generally adopted the Byzantine pronoia as the fundamental source for managing land and allegiance, becoming the timar system. There is a reason why they kept the name Konstantinniye for Istanbul right until the modern period, after all - the legacy of Constantine weighed heavy on the Ottomans, although they also saw themselves as inheritors of the Anatolian ghazi tradition, and, after the 16th century, of the caliphate as well. There is some indication that court culture had profound Byzantine heritage as well, particularly in the varied processions of guilds in front of the sultan during different festivals.
Sources: Runciman's Fall of Constantinople, Quataert's Ottoman Empire, Faroqhi's Subjects of the Sultan
Mehmet wasn't quite the only one who used the title -- Suleyman the Magnificent also drew on the title, and it was a key bone of ideological contention between Suleyman and the Holy Roman Emperor at the time, Charles V.
As Leslie Peirce notes:
Charles' elaborate investiture in 1530 as Holy Roman Emperor compelled Suleyman to react with his own staging of imperial glory, the sultan appeared to gain the upper hand in 1597 when Charles and his brother the Archduke Ferdinand sued for peace -- Charles was demoted to "the king of Spain" in the language of the treaty and Ferdinand humiliated by the imposition of an annual tribute ... Suleyman now had greater license to style himself the sole Roman emperor, a pretention that perhaps inspired the plans for the great mosque whose first stone the sultan laid in 1550. (Re-Reading the Black Legend: The Discourses of Religious and Racial Difference in the Renaissance Empires, p. 32)
And again:
... the Ottoman heartland was a Roman landscape: the capital in official parlance was the Arabized "Konstantiniyya" ... This interleafing of dynastic and territorial identities made the point that the Roman empire of the Ottomans, like its European counterpart, was holy and Roman-- it was, finally, Muslim. (ibid, p. 36)
Kaya Şahin also discusses this struggle between "two Roman Emperors" in Empire and Power in the Reign of Süleyman, pp. 81-87, saying that "the Ottomans were ... intent on contesting Charles' titles on the battlefield" and this "ideological dimension determined the tenor of the "German" campaign from the very beginning" (p. 82).
It's correct to say that the Ottomans didn't really press the title from this point on, though. I do think it's worth pointing out, however, that the Roman identity formed some portion of the Ottomans' imperial legitimacy, and the Ottomans certainly inherited many of the policies assumed and challenges faced by the eastern Romans.