When we see the comment that someone “began to be old” in the Book of Mormon, we are being notified that they are about to die. Thus, in verse 9 Nephi begins to be old, and in verse 12 “it came to pass that Nephi died.”
The rest of the information Jacob gives is part of the command to only touch lightly on history. This is important history, even more important because we have lost the 116 pages that would have given us more of this history. We knew that Nephi was a king, and here Jacob notes that, as is typical, he was followed by kings. Jacob does not say that the next king was Nephi’s son, but that is the logical assumption.
What Jacob does state is that when the next king ascended to the throne, he was called Nephi. What is not clear is whether this is a name change, or whether the designation of Nephi became a term for the king. Julius Caesar was the emperor, and subsequent emperors bore the title Caesar along with their own name.
It is also interesting that Jacob tells us that Nephi “wielded the sword of Laban” in defense of his people. In Words of Mormon 1:3 we will learn that king Benjamin also wielded the sword of Laban in defense of his people. After that time, we see the sword as a sacred relic, but not necessarily as functional in battle. Of course, even in these cases, it may have been more symbolic. With a hilt of pure gold, Laban’s sword was probably more ceremonial than military.