It also could be that the Basillisk has some sort of genetic memory (or DNA-based cognition ala the Super Happies!) such that the monster in the book is not the original monster but rather a great-great grandwhelp of the original monster. This would allow any heirs to kill their specific monster while the line (and thus memories) are preserved.

(This is of course all predicated on Slytherin realizing that his descendents may be nasty enough to keep knowledge from others by any means possible).