There isn't one person who named the days of the week as we know them today, but some of the first records of the names can be traced back to ancient Roman times.
At the time, in addition to the Romans, there were tribes all over Europe in what we now consider Germany, Norway, and England. All of these tribes had their own words for the days of the week, which they often named to honor different gods, and when these tribes met they would combine languages. When these new languages finally reached Rome, the names for the days of the week were among some of the words the Romans adopted!
The individual days' names are often thought to be traced back to these specific origins:
Monday - From "Day of the Moon." Also sometimes thought to reference the Germanic god of the moon, Mani.
Tuesday - Named after Tyr (sometimes spelled "Twi"), the Norse god of war.
Wednesday - Named after Odin (also referred to as Woden), the chief god of the Germanic people.
Thursday - Named after Thor, the Norse god of thunder.
Friday - Named after Frigg, a Germanic goddess and Odin's wife.
Saturday - Named after Saturn, the Roman god of harvest.
Sunday - From "Day of the Sun."