The exact meaning of an Enfant de Marie (Fr. Child of Mary) is unclear.
Possibly, it is related to an order known as the Congregation of the Children of Mary which was founded in Castelnaudary in the South of France by a Father Dulignan, sometime between 1830 and 1842. This order is what is known as a Third Order (one whose members, known as Tertiaries, live normally in the outside world, rather than being part of an institutionalised religious community).
The term Enfant de Marie is also linked to an organisation formed on the basis of a vision experienced by St. Catharine Laboure in 1835, which was especially popular during the mid-20th Century. After a six month period of 'aspirancy', its young women members earn the right to wear a blue cape which they relinquish on marriage.
Does anyone have any more information on this?