Aramaic was an important substrate in Palestine, but in the Byzantine empire Greek had a huge importance. In former Sassanid Empire and Byzantine lands, the Arab conquerors retained the bureaucracy structure and language for a period of time and only very gradually it was abandoned for Arabic. Btw Semitic speakers (Aramaic in the area you asked about, but "Sudarabic", sorry I don't know the name in English, in Arabia Felix in Yemen too)had a easier time abandoning their language for Arabic, as it was linguistically closer to them than for Persian speakers. Keep in mind that transition happened during centuries as more people converted and rule strengthened. Languages became substrata for dialects gradually and became forgotten or reduced to just minority languages for specific or isolated communities. In Egypt Coptic became truly to decline during middle age famines, when whole groups changed religion en masse in order to profit from the more advantaged status of Muslim.

By the seventh century Syriac, a dialect of Aramaic, had become the lingua franca of much of the Near East between Palestine and Iraq. This is very visible in the the literary texts we have, the vast majority of which were of Christian provenance (we basically have no contemporary Persian texts). In reality of course people spoke all kinds of other languages/dialects, but Syriac was becoming the foremost written dialect amongst native-Aramaic speakers, who dominated this region in terms of numbers. Administratively, people in the Roman Empire continued to use Greek and the Sassanians Persian, whilst on the borderlands with Arabia plenty of people would have spoken Arabic, especially given the power and influence of the Arab confederations guarding imperial borders in the sixth century. This is widely attested to in inscriptions (at least on the Roman side of the border) and I have read about several tri-lingual inscriptions in Syriac, Arabic and Greek, the three most useful written languages in the region. There is unfortunately no way for us to calculate the exact linguistic make-up of the region, though given the ethnic and religious diversity found within the region, I would suggest that it was very mixed.

For further reading, particularly on the rise of Syriac, anything by Sebastian Brock would be very useful as he is the most influential scholar on this topic. More broadly, the development of Syriac and its role in creating a Syrian identity was expertly dealt with by Philip Wood in his ‘We have no king but Christ’: Christian Political Thought in Greater Syria on the Eve of the Arab Conquest (2010).