Great question. Every Greek polis was a combination of the urbanized city proper, the astu, and the attached agricultural countryside, the chora under polis control. Syracusa, like Athens, had an enormous chora which was able to support a large population. The Corinthian "settlers" seem to have enjoyed a (mostly) productive relationship with the native inhabitants of Sicily from the get-go, enabling the rapid establishment of secondary settlements at Heloros, Akrai, Kasmene, and Kamarina, all within about a century of foundation. The long and short of it: the early city managed to quickly control a vast and resource-rich territory, enabling population growth, agricultural expansion, and the wealth inherent therein. Most other Greek citystates had a much more limited chora, with Attica, the Corinthia, and Syracusa's chora being among the largest.
Like the "mother city" Corinth, Syracuse was well situated to dominate nearby sea routes. It is very likely that the Phoenicians had already been at Syracuse before the Greeks or in cooperation with them, probably on Ortygia island (the urban core of the old city). There is "red" pottery found there, now moldering in a crate in the basement of the museum, which is almost certainly Phoenician redslip, about which we knew virtually nothing before the 1960s. John Boardman died before getting around to it, and to my knowledge no one has showed interest in it since (there was a time when I had interest in such a project, but that ship has sailed, all puns intended). The name "Syracusa" is probably a Phoenician phrase which means "seagull rock." Anyway, I digress. The Phoenicians (maybe, probably) and then the Greeks rightfully saw Syracuse as well-situated to oversee traffic between the Western and Eastern Mediterranean, especially the narrow Straits headed up to the western coast of Italia and the lucrative Archaic markets of the Etruscans there. Syracuse and Carthage were, in some ways, in competition to be the "gateway to the West," but it was marginally more simple to sail to Syracuse from Greece than to sail to Carthage in the Archaic period. The harbor at Syracuse was also extensive. Syracuse maintained a powerful navy which went relatively unknown to history until its clashes with the Athenians in the harbor in 415 BCE during Athens' ill-fated invasion.
So: like Corinth and Athens, Syracuse enjoyed the double benefit of a large and productive chora and also a powerful maritime position as a gateway between West and East. And also like Corinth, Syracuse very early on expanded (diversified) its "portfolio" with secondary dependent settlements. In 480 BCE, at the battle of Himera, the western Greeks, led by Syracuse, successfully repelled Carthaginian attempts to check the expansion of Greek power, effectively asserting a status quo of Syracuse as a regional power in the central and western Med. In this same year, by the way, the mainland Greeks were facing the invasion of the Persian Xerxes. It was a momentous couple of years for the Greeks.
Pretty much everything I've mentioned so far has been geographical in nature. On this sub I typically argue strongly against geographic determinism: it takes more than good geography to make a successful ancient city, and many ancient cities existed and thrived despite their terrible geographical constraints (the classic example being Rome). There are other, less obvious factors at work with Syracusan success in Sicily like other settlements, which either remained small or failed (Leontini, for instance, or Megara Hyblaia). One such is the iron-fisted rule of a very narrow oligarchy of original settler elite clans during the Archaic period, whose governance proved to be effective. Another is Syracusan relations with the native interior of Sicily, about which topic I am not very knowledgable.
There is a relatively new book on Sicily by Richard Evans: Ancient Syracuse : from foundation to fourth century collapse (Routledge 2016). He hits all the early high points and reviews what we know and don't know vs what our sources tell us, sometimes unreliably. He covers Syracusa's tumultuous 5th century, the institution of the democracy, the Athenian invasion, and down to the 4th century. It's a really good new basic monograph on the history of the city.
Edit: For Phoenician pottery typology:
Schubart 2002, "The Phoenician Settlement of the 8th Century in Morro de Mezquitilla (Algarrobo, Málaga)" in The Phoenicians in Spain: An Archaeological Review of the Eighth-Sixth Centuries B.C.E., ed. Bierling (Penn State Press).
Bikai, P.M. 1978. “The Late Phoenician Pottery Complex and Chronology.” BASOR 229, 47-55.
Maass-Lindemann, G. 1985. “Vasos fenicios de los siglos VII-VI en España.” Aula Orientalis 3, 227-239.
Maass-Lindemann, G. 1990. “Orientalische Importe vom Morro de Mezquitilla.” Madrider Mitteilungen 31, 169-177.