As it was the ancient heartland of the empire I imagine that it would be considered special to the medieval Romans. Obviously after the Arab conquests the Romans were never again in a position to retake the peninsula, but if the stars had aligned the right way would retaking Italy have been a higher priority than say, Syria?
Hey there! I kinda touch on this question in this post of mine here. The time period I was writing about in that post was during a period when the Byzantine empire did still hold some portions of southern Italy, namely Apulia and Calabria. They retained control of these territories until the 11 century. So to answer your question, yes Italy still held quite the significance to the empire before they lost it fully. They viewed the peninsula as Roman, which was actually the significant driving force for Justinian as he launched the campaign to recapture it after the fall of the Western Roman Empire.
Ok so bare with me because im about to go on a HUGE tangent.
It is theorized that even before Justinian's reconquering, that Italy was better off. After the collapse of Western Roman hegemony in Italy, the peninsula later came under the rule of the Ostrogoths ruled by their king Theodoric the Great.
Theodoric sought to preserve and even build upon the Roman way of life by keeping Roman traditions such as the Senate of Rome active. His reign is seen as one of the most peaceful times in medieval Italian history that included vast building projects throughout the whole peninsula but concentrated more in the capital of Ravenna.
But back to my main point, under Theodoric a precedent was set of paying homage to the remaining Roman Empire of the east. In this fashion the Ostrogoth kingdom was seen as a vassal to that of the Byzantine empire. Having been raised in Constantinople as a hostage/ward in order to secure the obedience of the Ostrogoth king, Theodoric was raised around Roman tradition and customs. This is how he came to revere the Roman way of life and why he went out of his way to continue promoting this idea of Romanitas under his reign in Italy after the fall of the Western portion of the Roman Empire. So even after the fall, Roman life on the peninsula continued on almost untouched with even the Eastern Roman Empire's hands firmly(slightly) on the reigns.
But sadly after Theodoric's death things took a turn for the worse. His grandson took the throne under the regency of his mother, Amalsuntha. She was adamant about educating him in the Roman tradition just as Theodoric had been. But after his death in 534, the kingship was offered to Amalasuntha's cousin who later had her captured and killed which prompted the Gothic Wars of Justinian which subsequently ended with the Italian Peninsula under full Byzantine control.