Where is the dad's family in all of this? If the mom can't take care of her son, the first person whose responsibility it is to care for the child is the father. And if he can't or won't, both sides of the family should be looked to for support, not just yours.
One thing you might suggest is that, as you don't have the ability or inclination to take in a child, maybe the family can all chip in to ease the financial hardship on whoever becomes primary caregiver? Between multiple cousins, and I assume the parents of you and those cousins, plus the husband's family, providing food and clothing for a child could split out (evenly) as a relatively minor cost for all each month.
(And I suppose, depending on how traditional Catholic your family actually is, you could always suggest sending him to a monastery. )