The names of the week were originally Roman according to the Oxford English Dictionary:
The Latin days of the week in imperial Rome were named after the planets, which in turn were named after gods (see discussion at week n.). In most cases the Germanic names have substituted for the Roman god's name that of a comparable one from the Germanic pantheon, but in the case of Saturday, the Roman name was retained and borrowed.
For example, Tuesday was originally named after Mars, but then switched to Tiw:
Originally cognate with or formed similarly to Old Frisian tīesdei , Old High German ziestag , Middle High German zīstag (German regional (south-eastern) Ziestag , (south-eastern and Swiss) Ziestig ) < the genitive of the Germanic base of the name of (the god) Tiw (see note) + the Germanic base of day n., after post-classical Latin dies Martis day of (the planet) Mars (4th cent. but probably earlier; frequently from c1135 in British sources; compare classical Latin Martis diēs
When the days were renamed, Mars was equivalent to Tiig (Tiw):
eOE Corpus Gloss. 76/2 Mars, martis, tiig.
The OED also notes that the Germanic people would have originally used the Roman names, but then subsequently changed them (by Old English, Mars didn't factor into Tuesday at all):
The English names, Sunday, Monday, etc., belong to an astrological week which, quite independently of the Jewish–Christian week, arose from the practice of assigning the successive hours to the seven planets in the order of their distance, and then naming each whole day (of 24 hours) from the planet supposed to rule its first hour. The planetary names, Dies Solis, Dies Lunæ, Dies Martis, etc., came into common use in the Roman empire, and were adopted in translated form by the English (before they came to Britain) and other Germanic peoples; the names Mars, Mercurius, etc., being apprehended as names of Roman gods, were rendered by the names of the Teutonic deities supposed to correspond to these
There is no further note on why Saturday was allowed to remain a reference to Saturn, however.
Best Answer
Between the 1st and 3rd centuries, the Roman Empire gradually replaced their previous system with a 7-day week with each day named after the planets of Hellenistic Astrology: The Sun, The Moon, Mercury, Venus, Mars, Jupiter and Saturn.
When the Germanic culture came in contact with the Romans, there was a practice where Germanic deities were identified with Roman equivalents. This process was known as Interpretatio germanica.
The Germanic peoples adapted the 7-day week system from the Romans and substituted the Roman gods with Germanic equivalents, except for Saturday, which retained the Roman god's name. This process happened after 200 AD but before the introduction of Christianity to Germanic peoples during the 6th to 7th century.
The answer is likely cultural pride. It's likely the Roman system was adapted for reasons of trade, convenience or war — but would you want to retain the names of the gods of your neighbours/enemies in the days of the week?
Mēnô was the Germanic moon deity (alternate spellings include Máni, Mōna, Māno,) so that's why dies Lūnae became Mōnandæg in Anglo-Saxon.
Why was Sæturnesdæg the only day that retained the Latin god's name? Probably because there was no Germanic god associated with Saturn.