Where did the belief about monks' mothers going to heaven come from? Some people believe that in Buddhism when the mother of a monk dies she would hold on to the monk's robe and go to heaven. 

I never read the Buddha say anything like that, but of course I haven't read everything the blessed one said. It makes me wonder because in Buddhism there are no shortcuts and it may look like one.

I've heard this belief exists in Thailand, does anyone have more information on that, like did the Buddha ever suggested it?
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