Where did the belief about monks' mothers going to heaven come from?

Householder Konrad, Interested,

in the case a mother gives her most beloved, let it go forth, that being a very high generosity, it is out of this reason, that she gains heavenly existence. Generosity (letting go of objects of the five senses, matter) is the cause of gaining heavenly existence (sugati realms).

In the case a mother rejoices this the gain of higher for her relative, mudita, such non-stinginess and joy of gains of others, is the case of gaining existence in the Brahma realms.

In the case a mother can at least mentally follow the path of leaving home into higher existence and happiness, such being the cause, seeing even the Dangers in the world, is the cause to gain existence in the Arupa-heavens.

If a mother follows the son's way of practice, goes for refuge into the Tripple Gems as well, such as the parents of Yasa, meeting the Buddha and his teachings, by the fact of entering the stream, on hearing it, she is destination to be no more able to fall of and as most Sotapannas, will gain heavenly realms.

Mv I 07: Pabbajjākathā — The Discussion of Going-forth [Ven. Khematto]. Yassa, a son of a money lender, disgusted of sense pleasure met the Buddha in the forest. After having been arisen to the highest Dhamma, his parents, on search for him, became, with his former wife, the first three-fold lay followers.

A very famous and inspiring story, next to that of the Buddhas Mother, Mahāmāyā, is the story of the mother of Ven. Sariputta Maha Thera, late then Upasika, Rūpasārī.

...Both she and her husband were unbelievers, and she was very sad when, one after another, her children, giving up wealth worth eighty crores, joined the Order. She wished to keep at least the youngest of the boys, Revata, for herself, and had him married at the age of seven, but her plot miscarried (See Revata). This embittered her against the monks, and, though she gave them alms when they came to the house, she blamed them for having enticed her children away...

This all being the reason, being the cause, having gained relation the family of the Noble Ones, is why it is said that the monk's mother (as she has to agree that one could become) is destined to heaven.

(Note that this Gift of Dhamma is not meant for trade, exchange, stacks, entertainment and akusala deeds, but as a share of merits and continue such for release)

This doesn't answer your question directly, but it's good contextual information I think. From the book "Women in Buddhism - Questions and Answers" by Chatsumarn Kabilsingh Ph.D.

An important point that Thai women put much emphasis on ordination of their sons is because they themselves have no opportunity to be ordained, so they depend totally on their sons to bring them this highest form of merit.

Also

The emphasis on the value of ordaining a son is a very highly recognised social value particularly in the village. Ordination is the only time that the mother is given highest honour as she actually is allowed to lead the precession, a place of honour to walk in front of the would-be monk holding the robe in her hands. Normally it is the men who would take the lead in all rituals in connection with the monks and the temples.

This is in a section of the book asking the question why some (poor) families would sell their daughters (into work as a sex worker) in order to get up front money to provide an elaborate and expensive ceremony to mark the ordination of their sons. It's a very eye opening discussion and gives insight into just how much importance is placed upon having a son ordain. Within this context it's easier to see how such beliefs arise.

This belief comes from a popular Chinese Buddhist tale in Mulian saves his mother from Hell.

Mulian Saves His Mother From Hell is a popular Chinese Buddhist tale originating in the 3rd century CE, inspired by tales from India of Maudgalyayana, who becomes Mulian in the Chinese stories. Mulian, a virtuous monk, seeks the help of the Buddha to rescue his mother, who has been condemned to the lowest and most painful purgatory in karmic retribution for her transgressions. Mulian cannot rescue her by his individual effort, however, but is instructed by the Buddha to offer food and gifts to monks and monasteries on the 15th day of the 7th lunar month, which established the Ghost Festival (Chinese: 鬼 節; pinyin: guijie). The monk's devotion to his mother reassured Chinese that Buddhism did not undermine Confucian filial piety and helped to make Buddhism into a Chinese religion.

The story developed many variations and appeared in many forms..

There are many variations and from that comes the simple belief that if a filial son could descend to Avici Hell and save his mother, a filial son will will always saves his mother especially if he is a monk. In Confucianism filial duty is inculcated and taken for granted

[This is not a direct answer, but comment space is too limited]

This kind of belief in short cuts to salvation appear to be a later cultural addition to Buddhism, though sending metta to ancestors is a well recognized aspect of Buddhism.

Texts like the Surangama Sutra, itself a later addition, says the Buddha participated in some such practices of saving ancestors from hells.

Then King Prasenajit, for the sake of his father, the late king, arranged on the day of mourning a vegetarian feast and invited the Buddha to the side rooms of the palace. He welcomed the Tathagata in person with a vast array of superb delicacies of unsurpassed wonderful flavors and himself invited the great Bodhisattvas. 1:121 [Shuranagama Sutra]

Acharya S.N. Goenka narrates a story that seems to imply nothing can improve one's lot except one's own merits. I can't find a source for this story independent of Goenkaji.

Youtube-(4 minutes audio) | (Text version-Google Books)

Pre-dating the Buddha, in Hindu brahmin culture the parents of a monk were guaranteed to go to heaven. This maybe a convenient form of consolation to a family that loses its healthy son who can plough the fields and serve them in old age. Promising rewards in the after life are the stock in trade of any religion to make people do something unpleasant right now.

Buddha was quite the exception to insist on dharma being good in the beginning, good in the middle, good in the end.

While dharma and meditation may help the individual, it doesn't directly help the family that has to accept the loss of a son. This is also why monasteries began taking in children at an early age - this made the loss acceptable to the family, since it hasn't spent a whole lot of time and money raising the son.