The dilemma is that my 19-yr old cousin was named after him
   too. (He actually goes by Henry, not Hank.) So the question we cant seem
   to figure out is whether its ok to name him after my grandfather, or
   whether the custom of not naming a baby after a living relative would
   apply. Obviously, we would be naming him after my grandfather, not my
   cousin, but...I just don't know. I have read conflicting opinions on
   this, and my mother said that her father actually had several first
   cousins with the same name, as they were all named after the same,
   deceased great-grandfather or grandmother, so she doesn't think it's a
   problem. My cousin also said it would be fine. That all said, I would
   love to know what others think.

If your families have no problems with it and your cousin especially is
fine with it, it seems okay to me to name more than one living child after
the same dead relative.  They'll be called different things and have
different middle names at least.

My 4 month old was named after my grandma, who died in 1995.  It was an
easy decision though in part because no one else in the family is a Miriam.

           The dilemma is that my 19-yr old cousin was named after him
   too. (He actually goes by Henry, not Hank.) So the question we cant seem
   to figure out is whether its ok to name him after my grandfather, or
   whether the custom of not naming a baby after a living relative would
   apply. Obviously, we would be naming him after my grandfather, not my
   cousin, but...I just don't know. I have read conflicting opinions on
   this, and my mother said that her father actually had several first
   cousins with the same name, as they were all named after the same,
   deceased great-grandfather or grandmother, so she doesn't think it's a
   problem. My cousin also said it would be fine. That all said, I would
   love to know what others think.

If your families have no problems with it and your cousin especially is
fine with it, it seems okay to me to name more than one living child after
the same dead relative.  They'll be called different things and have
different middle names at least.

My 4 month old was named after my grandma, who died in 1995.  It was an
easy decision though in part because no one else in the family is a Miriam.