Flower of Maryam: A Myth
You are right to think that using the flower of Maryam for labour is a myth, because the opening of the flower beside the pregnant woman who is due to give birth has nothing to do with making childbirth easier.
Kaff Maryam: A Treatment
Perhaps the family is confused and is mixing between “ward Maryam” and “kaff Maryam (lit. the hand of Maryam).” The former is what is mentioned in the question, which is a myth, as noted above. The plant called “kaff Maryam” is the one that has to do with childbirth, as it is proven that it helps a woman in childbirth.
The plant called “kaff Maryam” is known scientifically by the name Anastatica hierochuntica. It is used to treat colds in children, by steaming, and an infusion of this plant is drunk to make childbirth easy, if it is difficult.
Dr. Jabir ibn Salim al-Qahtani – who is a specialist in herbal medicine – said:
“Kaff Maryam” is a plant which resembles a clenched fist, and is used when a woman feels that labour has started; it is ground and soaked in a cup of cold water for four hours, then it is strained and the water is drunk only once.” (Jaridat ar-Riyadh (newspaper), 28 Rabi‘ al-Awwal 1428 AH/ 16 April 2007 CE, issue no. 14175)
It is believed that the name of the plant comes from a fable which claims that Maryam (peace be upon her) grasped it in her clenched hand whilst giving birth.
You can see a picture of the plant here:
http://www.khayma.com/hawaj/HERBPIC2/KFMRYM.JPG
And Allah knows best.
For more, please see these answers: 119767 , 161204 , 10538 ,and 110531