Why were the planets named after the gods the ancients named them after? More specifically, why was Saturn given his own planet instead of a more... prominent god?

Saturn was absolutely a prominent god (or really, Titan).

In Greek myth, Kronos played a very important role in the succession myth--Zeus supplanted Kronos, and thus established himself as king of the gods. The Titanomachy was a very important myth, and gets a major treatment in Hesiod. I remember in one of my mythology classes, I read somewhere that the succession myth was meant to signify the transfer of power from generation to generation; I'll see if I can dig up a source. It is also one of the oldest myths of the ancient Mediterranean, and there are parallels in near-eastern mythology.

So, in Greek myth, Kronos was important for two reasons: 1) he was the father of Zeus. 2) He was king of the Titans and was overthrown in the succession myth.

In Roman myth, Saturn becomes even more important. Saturn is believed to have been an Etruscan agricultural god originally, but when the interpretatio Romana occurs, he became associated with the Greek Titan Kronos. The Romans believed that Saturn, after being overthrown, ruled over Italy in a Golden Age, and had his citadel at the site of Rome. According to folk etymology, Latium, the region in which Rome was located, took its name from the fact that it was where Saturn hid away (latere). One of the major Roman festivals was Saturnalia, which occurred for about a week every December.

Saturn occurs frequently in Roman culture. They named a day on their market calendar after him (dies Saturni, which became our Saturday) as early as the 1st century B.C., as attested in Tibullus I.3.18:

Saturni sacram me tenuisse diem.

"The sacred day of Saturn held me"

See, also, Vergil's version of the Saturn story, narrated by Evander to Aeneas as he tours the future site of Rome. Here, Saturn plays the role of a sort of culture hero, giving laws, culture, and agriculture to the people of Italy (Aeneid VIII:314-332):

‘The local Nymphs and Fauns once lived in these groves, and a race of men born of trees with tough timber, who had no laws or culture, and didn’t know how to yoke oxen or gather wealth, or lay aside a store, but the branches fed them, and the hunter’s wild fare. Saturn was the first to come down from heavenly Olympus, fleeing Jove’s weapons, and exiled from his lost realm. He gathered together the untaught race, scattered among the hills, and gave them laws, and chose to call it Latium, from latere, ‘to hide’, since he had hidden in safety on these shores. Under his reign was the Golden Age men speak of: in such tranquil peace did he rule the nations, until little by little an inferior, tarnished age succeeded, with war’s madness, and desire for possessions. Then the Ausonian bands came, and the Siconian tribes, while Saturn’s land of Latium often laid aside her name: then the kings, and savage Thybris, of vast bulk, after whom we Italians call our river by the name of Tiber: the ancient Albula has lost her true name.

As you can see from this, Saturn played a major role in both Greek and Roman culture, which is why he merited having a planet named after him.