Why does "Not in a month of Sundays" mean "It won't happen"?

I'm not English so I find it hard to guess the reason why "Not in a month of Sundays" means "It won't happen" or "A long period of time".

I find the meaning weird. Can anybody explain to me why it means this? My question is why.

Full context (A children's literature by Jacqueline Wilson):

But my mum's coming to see me at Christmas. She is. I just know she is. 'Your mum's not coming to see you in a month of Sundays,' said Justine Littlewood. 'Your mum's never ever coming back because she doesn't want anything to do with an ugly manky bad-mouthed stupid show-off who wets the bed every ni—' "*

Solution 1:

According to the following source it probably derives from the Christian concept of Sunday as a "day of rest" from which the notion of a very long time:

• The expression is said to mean 30/31 weeks (the amount of time it takes a month of Sundays to pass) and has is believed to have origins from the Christian Holy Day of Sunday, the Sabbath. This day was a “day of rest” and was a long, solemn day devoid of amusement. Activities were even regulated on Sunday by law at times and therefore Sunday could seem long and tiresome (out of boredom)… therefore a month of Sundays could feel like an eternity. It is also sometimes used to denote something that will never happen.

The Oxford English Dictionary cites the first printed use of the phrase from 1759:

• “The commander..swore he should dance to the second part of the same tune, for a month of Sundays.” H. MURRAY Life & Real Adventures Hamilton Murray I. x. 121

NOTE: There are some variations on this, such as: Week of Sundays, Week of Saturdays, etc.

(makingheadsortailsofidioms.com)

Solution 2:

Two of the first citations (1841 and 1849) I find [in Google Books] for it are

I would give you a month of Sundays and you could not guess; so I will tell you.

He would not guess it in "a month of Sundays," neither shall we enlighten him.

My guess is that it started with expressions much like these.

Since Sunday is a day where you don't have to work, a month of Sundays is a whole month where you don't have to work. Thus, this means that he could have a whole month where he could devote all of his time to guessing this answer, and he still couldn't do it.

And a similar usage from 1835:

Your money I'll hide so that, if they were to search for a month of Sundays, by Jasus ! they'd never find it.

Solution 3:

There's a similar idiom in French:

"La semaine des quatres jeudis".

A literal translation would be : "The week with 4 thursdays". It means some time that you might wish for (there was no school on thursdays until the 1970s) but will never happen : no week will ever have 4 thursdays, just like no month will ever be full of Sundays.

It doesn't mean that the event is unlikely or might happen in a very long time. It will never happen.

This expression appeared in Rabelais' work 5 centuries ago. It is well known in French, and it might have inspired "Never in a month of Sundays" since the English idiom appeared much later.