Judging from the volume and innovative quality of source material in the Babylonian Talmud in contrast with the rest of the rabbinic corpora, the Babylonian contribution would appear to have been quite decisive in shaping the holiday observances in the direction of emphasizing the kindling of light(s). We see this right away from a couple of factors.
Firstly, the Babylonian Talmud chose to place its detailed discourse on Hanukkah in tractate Shabbat, as a digression within a chapter that discusses the suitable oils and wicks for lighting Sabbath candles. This very location, instead of one of the loci where Hanukkah is already discussed in the tannaitic corpora, as one might have expected, emphasizes the candle(s) as the central aspect of the holiday.
Secondly, although six traditions in the Jerusalem Talmud (henceforth: PT) refer to kindling lights on Hanukkah, a closer look suggests that there, too, the Babylonian input was predominant. Three of the six are based explicitly on cited Babylonian traditions. The remaining Palestinian traditions are not early—two of the three involve fourth century Palestinian rabbis. However, the quantity of such PT traditions is minute in relation to the Bavli’s treatment, and their content is also strikingly odd and lacking a broader contextual framework. Since the dated PT traditions that do not cite Babylonian rabbis are, in fact, chronologically later than those that do, this raises the possibility that even this PT material is ultimately an outgrowth of Babylonian deliberations.
Geoffrey Herman, “Religious Transformation Between East and West: Hanukkah in the Babylonian Talmud and Zoroastrianism”, in Religions and Trade: Religious Formation, Transformation and Cross-Cultural Exchange between East and West, ed. Peter Wick and Volker Rabens (Leiden & Boston: Brill, 2014), 266-267.