The papacy was a part of the Roman empire (apart from a 'barbarian' interlude between 476 and 536), so it's not that odd that quite a few pontiffs were originally from Syria. However, I suspect that you are asking specifically about the five Syrian popes that reigned (non-consecutively) from 685 to 741, which on the surface is very strange - after all, Syria had been lost to the Arabs in the 630s! I won't address the two Syrian popes before the seventh century, since I know nothing about either St Peter or Anicetus (there might be more, I'm relying on wikipedia for this), but the rise in eastern influence over the papacy in late antiquity is a fascinating topic, so I hope you don't mind me going on a bit of a rant here.
I think the first thing we have to note that is that imperial subjects were surprisingly mobile in this period. Armenian soldiers and officials for instance were posted throughout the empire, and we also have a record of a Lombard auxiliary serving in the east against the Persians. People within the church were no exception. Many future popes had for example first served as the papal ambassador to Constantinople, in the process meeting people from all kind of backgrounds. The future Pope Gregory the Great, perhaps the most famous of these apocrisarii, had for example met the Spanish churchman Leander of Seville in the imperial capital, whilst from his later letters it is also evident that he had developed warm friendships or at least respectful relationships with various eastern bishops and monks, all of which naturally facilitated by their underlings travelling back and forth across the Mediterranean carrying letters for their masters. More importantly, Gregory had also developed close links with various imperial officials. Strictly speaking imperial territories in Italy were ruled by the exarch of Ravenna, but the pope's influence was constantly growing and already in the sixth century he essentially looked after the city of Rome rather than the secular dux of the city. Again Gregory's letters provide the best example, as he repeatedly acquired grain for the inhabitants of Rome and got involved in various diplomatic dealings and/or incidents with the Lombards in order to defend it. Though the papacy was obviously a religious institution, for this period we should also see the figure of the pope as an imperial official.
Why is this important? Well, in the seventh century various crises in the east continuously drove refugees west. The extent of this is absolutely debatable and there is already evidence of easterners in the west before this, but some of the most prominent Greek writers of this period did flee west, most prominently the circle of John Moschos, Sophronius, and Maximus the Confessor. Moschos was the oldest and came from Cilicia, Sophronius from Syria, and Maximus was born in Palestine, yet all three probably fled together first to Egypt, then North Africa, before finally reaching Rome (they may have stayed in Constantinople on the way too), all in a bid to avoid the ravages of the war against Persia (603-628). They are the people we know the most about, but there are signs that there were already thriving communities of easterners in Rome, so they, as well as the many unnamed refugees we don't know about, were probably not doing anything too unusual. Various factors also made the west far more attractive than the east. Most obviously, the Persians had occupied the Levant after 614 and Egypt after 619. Anatolia was constantly raided and even Cyprus was perhaps under threat. The Balkans wasn't doing much better, since the Slavs and the Avars were on the ascendant, culminating in the Avar siege of Constantinople in 626. In that sense, fleeing to the relatively stable western provinces of North Africa and Italy was quite logical. This was helped by Emperor Heraclius' order to end the financial aid given to ecclesiastical refugees in Constantinople, which surely provided another push factor for many eastern churchmen.
So this was the situation just after the Persian war. Many people probably went back, most prominently Sophronius and Maximus the Confessor (Moschos had died in Rome), but within a few years Maximus was back in the west, this time fleeing from the encroaching Arab conquests. There are only vague hints for the existence of non-ecclesiastical refugees in this period, but I see no reason why we shouldn't expect them to behave the same way as many monks did. These monastic refugees were in fact very influential. To make a long story short, they opposed the doctrine of monotheletism promoted by Constantinople and they allied with the papacy to do so, an alliance that culminated in the rebellious Lateran Synod of 649 and the revolt of the exarch of Ravenna in 652, who perhaps had hoped to use this as an excuse to seize power in Constantinople. Alas this rebellion failed, but it demonstrates quite well the influence of easterners in Rome. The Acts of the Lateran Synod were for example first drafted in Greek, probably by Maximus himself, before being translated into Latin, whilst the flurry of letters and angry polemics written during this period amply demonstrate the intensity of the debate across the Mediterranean. There was for instance another Palestinian deacon who had repeatedly traveled to Rome to drum up support for these dissidents, as well as an ex-patriarch of Constantinople who went to Carthage to have a debate with Maximus before going up to Rome to ally himself with the pope. There was also a significant movement of secular individuals westward as well, most prominently after Constans II moved his court to Sicily in 662, which left behind hoards of seals on the island and is thus good evidence of many officials settling in Italy as well.
Essentially, there was a strong presence of easterners within Italy at this point, so the existence of Syrian popes a generation or so later should not be unexpected. Some historians, such as Andrew Ekonomou, suggest that this was due to the demographics of the local clergy having fundamentally changed in the mean-time due to the population movement of the previous decades, so much so that when papal elections happened easterners were more likely to be elected by their peers. This is I think far too simplified, plus, as Thomas Noble rightly noted, all these Syrian and Greek popes probably identified themselves as Romans anyway and so were not very different from the 'native' inhabitants of the city - the fact that they were born in different places or had Syrian parents therefore didn't mean very much for their electoral chances!
Whilst it is difficult to explain the string of Syrian popes, I am far more inclined to emphasise the role of their duties rather than the importance of their ethnicity. Rome in the seventh century was a hugely diverse place, full of visitors ranging from Anglo-Saxon kings travelling as pilgrims to devotees of Persian martyrs, a fact confirmed by the many actors during the monothelete controversy. In that sense, the fact that a number of popes were 'Syrian' was I think just a coincidence, a demonstration of just how colourful Rome was at that time and an indication of how well the Italian church integrated people of diverse origins, but not much more than that. Our only source, the Book of Pontiffs, regularly noted each pope's parentage and/or place of birth, but never commented on it beyond that, so I don't see why modern historians have to dig deeper. It was just a custom for contemporary chroniclers of the papacy to note the pope's origins and it is quite clear that this had little bearing on the papacy's policy. Indeed, two Syrian popes, Sergius I and Gregory III, resisted imperial pressure the most in this period. Surely they had thoroughly identified themselves as Romans and I think that's how we should view them as well.
As for why there were so many Syrian popes, rather than say Palestinian or Cilician popes, I'm afraid I can't explain why given my current understanding of the evidence, though it is worth noting that in other western sources 'Syrian' is sometimes used as a generic term for easterners. I would also like to note that more popes in this period were born in Greece or Sicily than there were Syrian popes (one of the Syrians popes was actually born in Sicily!), which to me implies that popes in this period were more likely to be chosen if they had some connection with the imperial apparatus, as Sicily was a province dominated by the imperial administration and was where Constans II was based between 662 and 668, whilst Greece was obviously closer to Constantinople (though also dominated by Slavs in this period, I need to do a bit more digging to see if any historians have explored why there were so many Greek popes from a Greece largely ruled by Slavs). This brings me back to the point I raised many paragraphs ago - the pope was also a secular official acting on behalf of the emperor, so it would have been quite useful for a pope to have some connections with the court or imperial bureaucrats already, not to mention being fluent in Greek. I am however also intrigued by the idea that there were Syrians who went west after the Arab conquests, since Pope Sissinus and Pope Constantine reigned in the eighth century but were also recorded as being born in Syria. Not really sure how that worked, since from other sources the flow of people from the caliphate to the Roman empire was quite small, so this is absolutely something I have to do more research on.
In short, I would suggest that there were a number of Syrian popes because: