What is the rule of the eldest?

if you're looking for the definition it's just another Word for "oldest" or "first born"

but the meaning of the title to the 2nd book (in my opinion) it means this: towards the end of the book in the chapter "inheritance" murtagh grabs zar'roc from eragon and says that he's the rightful owner of zar'roc by being the eldest child of morzan, not eragon, who is the youngest child of morzan