The origin of day is Old English dæg ("day" or "lifetime") from Proto-Germanic *dagaz, which derived from Proto-Indo-European *dhegh-. In English, the original meaning was "the daylight hours" and wasn't extended to "the 24-hour period" until late Anglo-Saxon times.
Despite their similarities, day is not considered a relative of Latin dies.
In Old English, the days of the week were:
Sunday: Sunnandæg - "The day of the Sun"
Monday: Monandæg - "The day of the Moon"
Tuesday: Tiwesdæg - "The day of Tiw (Tyr)"
Wednesday: Wodnesdæg - "The day of Woden (Odinn)"
Thursday: Þunresdæg - "The day of Thunor (Thor)"
Friday: Frigedæg - "The day of Frige, or love"
Saturday: Sæterndæg - "The day of Saturn"
In case you are unfamiliar with the pronunciation of the Old English alphabet, here's some help:
Æ/æ - sounds like 'a' in cat
Þ/þ - sounds like 'th' in thick
Ð/ð - sounds like 'th' in those
G/g - sounds like 'j' in jeans