There is a theory propounded that "One week's notice" or "Three weeks' notice" are the correct ways to write these phrases. This is based on the entirely erroneous notion that the expanded forms of the phrases, "Notice of one week" and "notice of three weeks", indicate possession.
It requires very little thought (though, it would appear, more than grammarians are capable of) to determine that the "of" indicates composition, rather than possession - as, for example, in "eyes of blue", "rod of iron", "cup of tea" or "chariots of fire". The propounders of the "week's/weeks'" theory, by their own logic, would refer to "blue's eyes", "iron's rod", "tea's cup" and "fire's chariot".
In fact, "weeks" is an adjective. There are plenty of cases where a noun, in being converted to an adjective, takes a different spelling. For example "mud" and "muddy" - "field of mud" = "muddy field", not "mud's field". Another example is "flax" and "flaxen".
Time intervals when used as adjectives have an internal consistency of spelling which is not necessarily consonant with their noun form. A break of five minutes, for example, is a "five minute break", a countdown of ten seconds is a "ten second countdown". Hence the adjectival forms of "second" and "minute" happen to be the same as the singular form of the nouns. But the obvious corollary to these, that the adjectival form of "week" is "weeks", is not only ignored but regularly contradicted.
The correct spelling of "weeks" in the phrase "one weeks notice" takes no apostrophe. Ignore the false ones who say otherwise.