The idea of a supreme creator god was already well established in Arabia before the advent of Islam. Most obviously, the Kabaa santuary in Mecca was dedicated to al-Lah, the "high one". What got Mohammad in trouble during the early years of Islam was not the veneration of al-Lah, but his insistence on abandoning the worship of all "lesser" divinities, particularly the goddesses Mecca, Lat, Uzza and Manat, who were referred to as the Daughters of al-Lah. (This also led to the infamous brouhaha over the Satanic Verses.)

There is a story handed down about Mohammad, as a young child, meeting a Christian monk named Bahira, who fortold that Mohammad would become a prophet. This story was included in Ibn Ishaq's 8th century biography of Mohammad. Ibn Ishaq also stated that Khadija, Mohammad's first (and at the time only) wife, had a Christian cousin. After various portentous miracles, this mystical cousin supposedly advised Khadija that "if this be true, O Khadija, then Muhammad is the prophet of his people. I know that a prophet is expected at this time." Even if many of these tales can be discounted as post-Islamic attempts to emphasize Islam's continuity with the Judeo-Christian tradition, they at least point to Christian and Jewish beliefs being present and familiar throughout Mecca and Arabia at the time.

After the arrival of Islam, its early history was full of interactions with Jewish and Christian communities. For instance, prior to the Hegira, there was an earlier migration in which a small group of Muslims fled persecution in Mecca by migrating to the Christian kingdom of Aksum in modern-day Ethiopia. And after the Hegira to Yathrib (Medina), the nascent Muslim community came into contact with three prominent Jewish tribes who were co-existing with the Arabs in the city. The Jewish tribes were included in the Constitution of Medina, committing them to mutual cooperation with the Arab and Muslim tribes of Medina. However, Mohammad would later have a major falling out with one of the Jewish tribes, the Banu Qurayza.

To provide another perspective to /u/cyd's more traditional account, I will argue that Arabia in the sixth and seventh centuries were very much an integrated part of the wider world of late antiquity. It is impossible to ascertain what the 'average' person thought, simply because the evidence is not there, but there is plenty of evidence that there were Christian and Jewish communities within the peninsula.

It is however better to begin by discussing Islam. You mentioned the question of whether Islam was a 'syncretic combination' of older beliefs. This is I think a misleading way to look at it. For one thing, we know very little for certain about early Islam and its beliefs (aside from the demonstrably early evidence from the Qur'an, but scholars have very divergent readings of this important text), so it is rather difficult to answer the question. Secondly, it is perhaps less the case that it was a faith that drew on different traditions, but instead a direct continuation of extant beliefs – this is a subtle difference, but syncretism implies that it was an uneasy amalgamation, when in reality the region already possessed a rich Abrahamic heritage and was constantly in dialogue with events in the largely monotheist world beyond Arabia. For more information, take a look at /u/CptBuck’s answer here, as well as my take on the historiography here.

References to Christians and Jews in Muslim traditions have already been mentioned, but they were written rather late and should not be seen as directly reflective of events in seventh-century Arabia. For instance, many of Ibn Ishaq’s accounts concerning Christians and Jews dealt with how pious Christians and Jews recognised Muhammad’s greatness, in some cases before he was even born – these are classic indicators of a sacral history, with the past subtly reshaped to present a narrative of Muhammad as a foretold prophet welcomed by learned individuals. These later sources no doubt preserved memories of an earlier world in which it was the norm to meet Christians and Jews, but we have to be careful when we use this evidence to discuss the earlier period with regards to specific details.

It is I think better to rely on the contemporary evidence from the sixth and seventh centuries. I provided an overview of the turmoil of the sixth century here, but in short, the peninsula already had a number of Abrahamic rulers. In the south, Yemen (or at least some of the elite) had converted to Judaism in the fourth century (or thereabouts), only to fall under Christian Ethiopian rule in the early sixth century. The locals revolted under a Jewish leader and initiated a pogrom of the Christians, which had enormous diplomatic implications, for the great powers of late antiquity, Rome and Persia, became involved via their clients. It would appear that the Persians threw their weight behind the Jewish ruler of Yemen, whilst Rome backed Ethiopia and backed their expedition to take back the region. In 525 this was successful and the Ethiopian strongman who was now in control, Abraha, began to campaign across Arabia to secure his position, perhaps even attacking the Hejaz on one occasion (a memory that can perhaps be tied to the Year of the Elephant in Muslim traditions). Decades later, around the 570s, the sons of Abraha were deposed by the Persians, supposedly at the instigation of local Jews according to later sources.

In the north, Arabs were just as immersed in the wider world. On the Roman side, there was the powerful Ghassanid confederation ruled by the Jafnids. They were Arab clients, but that did not prevent them from being powerful in their own right, intervening for example as the secular arbiter in disputes within eastern Christianity. On the Persian sides, there were the Lakhmids under the Nasrids. They were pagan Arabs, but they had also ruled Christians and were likewise integrated into the imperial system. Notably, the last Nasrid ruler had converted to ‘Nestorian’ Christianity, an event that was even hailed by a Christian historian in Antioch, despite them belonging to separate churches. This chief was soon deposed for disloyalty and we do not know what his successors believed, but in the context of Christian majorities in all the surrounding regions and how integrated Christians were within the Zoroastrian Persian empire, it is difficult to escape the conclusion that these Arabs in the imperial frontiers were very much aware of other faiths, regardless of their personal beliefs.

Now we can discuss the thorny question of Christian/Jewish influence on Islam. The answer: we don’t know, certainly nothing approaching a consensus. There is a recent tendency to see Islam as a late-antique religion and that it was very much in tune with religious and cultural developments elsewhere. Take for instance the position laid out by Peter Sarris in his Empires of Faith (2011), which synthesised the work of the historians Robert Hoyland and James Howard-Johnston:

The religion of Muhammad as it emerges from the earliest sources was a faith deeply engaged with wider debates and concerns that were coming to dominate the religious life of the Mediterranean world into which Arabia, from the fringes of Syria to the Yemeni littoral, was increasingly being drawn. The verses of the Qur’an reveal a preoccupation with the imminence of the apocalypse, and a determination to counter those who denied the existence of the afterlife, such as we have already encountered in the writings of Pope Gregory the Great. Like Gregory’s Book of Pastoral Care (or the Spiritual Meadow of the seventh-century Palestinian monk John Moschus, who wrote under papal patronage), its verses offer an intensified but pragmatic piety whereby the core asceticism of the ‘Holy Man’ was available to all. The vision of paradise contained in the work, and the rituals associated with prayer that we encounter in early Islam, moreover, bear a striking resemblance to those found in the early Christian Syriac ascetic tradition. The respect in the Qur’an for the Virgin Mary, but its denial of Jesus’ divinity and crucifixion, chime closely with shades of contemporary Christianity, both ‘orthodox’ and ‘heterodox’. Indeed, it is striking that a Christian heresy is precisely what the eighth-century Syrian theologian John of Damascus regarded Islam to be. What Muhammad’s austere monotheism cut through were the complexities and controversies of Christian Trinitarian and Christological doctrine. In that sense it was itself a fulsome response to contemporary Christian debate.

[...]

That Muhammad should have preached a doctrine of martyrdom and holy war at precisely the time that these selfsame concepts were featuring so prominently in the official pronouncements of the East Roman state is highly suggestive, and may indicate that Heraclian propaganda was percolating into the Arabian peninsula. The region around Mecca appears to have been one of the few parts of Arabia that had not come under direct Persian control by the late sixth and early seventh centuries, and thus it may have been something of a bastion of pro-Roman sentiment. Certainly, sympathy for Heraclius’ cause is strongly indicated by the Qur’anic verse on ‘the Greeks’ (al-Rum), possibly recording the Persian defeat of Roman forces near Bostra in 613.

Because of the nature of the historiography, we should not see this as the 'right' answer, nor indeed any other position presented by scholars working on this topic. The late Patricia Crone has for instance argued forcefully for the influence of a specific strand of Christianity, Jewish-Christianity, on Muslim beliefs, whilst if we follow the work of Gerald Hawting, then most people involved in Muhammad's life were already monotheists, not polytheists, so it is a bit of a moot question to ask about Muhammad's views of Abrahamic faiths, since he was already living in an Abrahamic world. We can also consider Fred Donner's thesis, which is that Muhammad created a reformist apocalyptic movement open to Abrahamic monotheists of all stripes, so again, Muhammad did not live in isolation, but was acutely aware of of the beliefs and needs of local Christians and Jews.

It is difficult to decide which of these ideas are more convincing (although I guess Crone's work is the easiest to disagree with), but this sort of thing is the norm for those who study early Islam. I therefore encourage you to be wary of any bold confident theories on this topic, but be more cautious and critical, regardless of the authority concerned. It is easy enough to look at the Qur'an, with its frequent mentions of Biblical prophets, and the Constitution of Medina, with its inclusion of Jews into Muhammad's growing polity, to extrapolate what Arabian society was at that time - but when we essentially only have these two sources as contemporary attestations to the nature of Islam or the Hejaz in the early seventh century, we should instead be more careful when studying these texts.

Personally, I feel that Muhammad's teachings must have constantly engaged with that of Christians and Jews, who were not strangers in Arabia, but equal members of the same Arabian world. These monotheists were no different from the more familiar pagan Meccan elite or the nomads of the desert, so it is a shame that it is only in recent decades that their influence has become the focus of scholarly investigation. With more time, I am confident that a new narrative can be written, but we are not quite there yet.