In the depth of my mind there is a connection to Mediaeval French (Marie) in some way, maybe, in the same way that Jehan(e) is related to Jean(ne). However it may be that someone I knew who lived in a Francophone country called her daughtes Jehane and Mahala.
Malie is a cognate of Marie - as is Mail(l)e (Mollie is also a diminutive/cognate of Magaret). The r and the l can be interchangeable dialectically in some some periods/geographical areas.
I was not aware of the meaning sick or weakly.