Hi Ross,
Nassa is Latin for basket, specifically a sort of woven basket used for
catching fish.  The family got the name from the reticulated pattern seen on
so many species, which resembles the weave of a basket.  In fact, Nassarius
were referred to as basket shells in some older books - you don't see that
name much these days.  The prefix Ilya- I believe means mud or muddy, which
would make Ilyanassa a mud-dwelling basket shell.
Paul M.