This syndrome is called Munchausen by Proxy (as mentioned by ab2 in the comments).
Its name is derived from the Munchausen syndrome where a patient feigns or exaggerates having diseases him/herself; Munchausen refers to the literature character Baron Munchausen, who was based on a real-life baron who exaggerated a lot of stories about his military time.
The 'by proxy' part indicates the person having the disorder is using the person in their care as a 'substitute' instead, for feigning (or actually inflicting) the disease.