It is a funny coincidence, but it's very unlikely that there is any connection to the solstice or its various holidays. Kislev is a lunar month that falls in wintertime, but the 25th of Kislev is not calibrated to the solstice. While this year Hanukkah falls after the solstice, that's kind of unusual. Most of the time Hanukkah begins in mid-December.

The Hebrew calendar is "calibrated" to the Julian calendar, which means it's drifting later 3 days every 400 years. Originally the calendar was based on observation--witnesses would observe the new moon, and travel to the head Rabbinic court, who would interview them and declare the beginning of the month (the lunar month is between 28 and 29 days, so the month wouldn't go on forever if it was cloudy). Depending on how the seasons were looking, they would decide whether a year had 12 or 13 months. Already when there were some "fudges" to make the calendar work out more easily, and it's possible that the calendar was unofficially "rigged" from an early date in some way. The current calendar started sometime in the early Middle Ages (or late antiquity). While it used a highly accurate calculation for how long the lunar month is (so the day the month starts has not drifted significantly as compared to the real variation of the lunar month), the calculation of the solar year to keep the calendar in-sync was not so accurate, since it used the Julian calendar (incidentally--all observances based on the solar year in Judaism are based on the Julian calendar).

It's difficult to say how the calendar would be running if it changed to the Gregorian calendar--you'd have to re-do the whole system. This is why, for example, the coincidence of Thanksgiving+Hanukkah a few years ago was interesting. Because of the calendar drifting back, Hanukkah used to be able to fall on Thanksgiving, but will not again unless the calendar drifts so far as to have Hanukkah go from winter to spring to fall, which will take thousands of years, if the calendar is not reformed before that point. While Hanukkah can fall on dates of Thanksgiving still, it's become rare enough that they never happen to coincide.

All this means that in ancient times, Hanukkah was actually even earlier, and less likely to coincide with the equinox. If the calendar was fixed at around the 8th century (which is around the latest that the Jewish calendar in its modern form could've been fixed), the median date wouldn't've been December 12ish, it would've been more like December 5ish. In fact, the solstice would've been an exceptionally late date! Before then when the calendar was fixed but not yet in its exact modern form, and it's hard to know what the year was "calibrated" to. The evidence seems to suggest that the lengths of months was "rigged" before the leap-year cycle, but it's possible that the leap years were calibrated to the Julian calendar from an early date, rather than using independent observation. If so, originally Hanukkah would've fallen in the timeframe corresponding to modern November half the time, nearly a month before, and falling on the solstice would've been a rarity. If the Rabbis had calibrated the Jewish calendar more accurately, we'd associate Hanukkah with Thanksgiving!

The only historical evidence regarding the date is basically the traditional story. Namely, that when the Maccabees rebelled against the Seleucid Empire, they reconquered the Temple and purified it on the 25th of Kislev. While it's possible this date was chosen for some ideological reason, there's no obvious reason that should be the case. The date isn't connected to the solstice, or anything really. The most likely scenario is that it simply was the date the historical event happened.