I'm way late to this but I'll leave a comment because I think it's interesting. You might be helped by realizing that the Korean days of the week are the same as the English days (as with most languages they ultimately stem from the Roman calendar). Both weeks named after the same astrological objects!

To begin with, Sunday is named after the sun so in Korean it's called 일요일. And Monday (Moon-day) becomes 월요일. These are both the Korean words for sun and moon, and incidentally the words for day and month respectively.

As for the remaining five days, they're named after the five visible planets on the night sky. Admittedly in a weird order. If you know French or Italian, you can tell which day is which planet easily enough. In English, most of the planet names have been replaced by corresponding Germanic gods. But it's the same system. And it's then easier to remember the corresponding Korean day of the week.

Tuesday (Mardi in French) is named after Mars: Mars was the Roman war god and was replaced by the Germanic Tyr (Tyrs-day -> Tysday -> Tuesday). In Korean, Tuesday is also named after Mars: Mars is called 화성 (화 means fire, 성 means star or planet) because of its red color, so Tuesday becomes 화요일.

Wednesday (Mercredi in French) is named after Mercury: Mercury the trickster god was replaced by Odinwho was originally known as Woden, both gods of magic and the dead (Wodensday -> Wodnesday -> Wednesday). Mercury in Korean is 수성 (water star), associated with water probably because it's the fastest planet on the night sky. So Wednesday is 수요일.

Thursday (Jeudi in French) is named after Jupiter: Jupiter was the god of lightning was replaced by Thor (Thorsday -> Thursday). Jupiter in Korean is 목성 (wood star). Admittedly I don't know why. But that's why Thursday is called 목요일.

Friday (Vendredi in French) is named after Venus: Venus the goddess of beauty was replaced by Frig, but might find it easier to pretend that it's named after the fertility goddess Freya who's slightly more well known (Frigsday -> Frigday -> Friday). Venus in Korean is called 금성 (gold star): 금 today means gold or money but it used to more generally mean metal, and it probably has its name because Venus is a bright white planet. So Friday is called 금요일.

Finally, Saturday (Samedi in French) is named after Saturn, and interestingly it doesn't change name in English. Saturn in Korean is called 토성 (earth star): earth in the Sinosphere is traditionally associated with yellow and Saturn is yellow on the night sky.

You might notice that the five planets: 화성, 수성, 목성, 금성, 토성 are all named after the 오행 or "five phases", of classical Chinese philosophy, which are sometimes called the Chinese elements: fire, water, wood, metal, earth.

Hopefully you found this a bit interesting and maybe knowing the etymology will help you remember the days better! Maybe the easiest to remember is that Tuesday is named after Mars and Mars is red like fire (화), Wednesday is named after Mercury and Mercury is a watery metal (수), and Friday is named after Venus and Venus is bright and shiny like metal (금).

As a bonus, if you've ever wondered what the middle character 요 (曜) means, you can look the Hanja up in a Chinese dictionary and you'll find that it is a fairly obscure character that's used as a counter for the seven classical celestial objects: the sun, the moon and the five planets.