Mosiah 1:11 - 1:18

Mosiah 1:11 And moreover, I shall give this people a name, that thereby they may be distinguished above all the people which the Lord God hath brought out of the land of Jerusalem; and this I do because they have been a diligent people in keeping the commandments of the Lord.

Not only will king Benjamin tell them who will succeed him but he will give them a name which is a reward for their successful efforts in keeping the commandments of the Lord. This name will give them special distinction from all other peoples that have been lead away from Jerusalem.
[Don’t quite understand his naming of a people. It maybe that they had not have a name such as LDS members who call themselves Latter-day Saints. They may have only been called Nephites which has no religious connotation like being called an American. Giving them a name with a religious connotation will hopefully cause them to think twice about doing wrong.]

Mosiah 1:12 And I give unto them a name that never shall be blotted out, except it be through transgression.

The only way these people would lose this name would be if they turned to evil continuously.
[I assume they would not like the name which has a religious connotation as it would label them being what they did not live as or would hold them up to ridicule from their fellow wicked friends such as we see in Lehi’s dream. It is like a wife losing the name of her husband through divorce. This giving of a name is a marriage between the people of king Benjamin and the Lord. If they do not live up to the commandments, they will be not be eligible to live with him and will be cast off.]

Mosiah 1:13 Yea, and moreover I say unto you, that if this highly favored people of the Lord should fall into transgression, and become a wicked and an adulterous people, that the Lord will deliver them up, that thereby they become weak like unto their brethren; and he will no more preserve them by his matchless and marvelous power, as he has hitherto preserved our fathers.

Therefore if the people of king Benjamin do not live up to every covenant they make with the Lord on this occasion which has brought them many blessing that they could not or did not earn, they will not receive anymore. If they become a people who think they are prospering because of their own abilities and who are not being loyal to their marriages or family responsibilities, then the Lord will withdraw his help and they will be on their own when confronting their enemies. It was only through the help of the Lord that they have been able to conquer the Lamanites up to this point. If the people think they can do it on their own they will find out to the contrary.
[It is hard for of us to think that a force unseen by us is what is protecting us from powers that seek to destroy us. We tend to think that we are strong by our wise decisions and preparations. However it is by the direct hand of the Lord that we are free from those who want to make slaves of us. Otherwise we are on our own and fall pray to whatever force or evil design that exists that wants to make us bow to their will.]

Mosiah 1:14 For I say unto you, that if he had not extended his arm in the preservation of our fathers they must have fallen into the hands of the Lamanites, and become victims to their hatred.

King Benjamin testifies that if the Lord had not directly intervened in the conflicts with the Lamanites over the years, they would have been destroyed by them.
[I think we need to have this same conviction with the growing the Church in our day. It has been by the hand of the Lord that we have not been destroyed by those regardless of their intentions or motivations.]

Mosiah 1:15 And it came to pass that after king Benjamin had made an end of these sayings to his son, that he gave him charge concerning all the affairs of the kingdom.

After this discourse to his son Mosiah concerning the importance of knowing that the Lord is their preserver not their own strength, king Benjamin give him a turn over and tells him what his duties will be as king.

Mosiah 1:16 And moreover, he also gave him charge concerning the records which were engraven on the plates of brass; and also the plates of Nephi; and also, the sword of Laban, and the ball or director, which led our fathers through the wilderness, which was prepared by the hand of the Lord that thereby they might be led, every one according to the heed and diligence which they gave unto him.

Mosiah is given stewardship over the brass plates, the records kept up to this point on the plates of Nephi, Laban’s sword, the ball that lead Lehi to the promised land.

Mosiah 1:17 Therefore, as they were unfaithful they did not prosper nor progress in their journey, but were driven back, and incurred the displeasure of God upon them; and therefore they were smitten with famine and sore afflictions, to stir them up in remembrance of their duty.

King Benjamin reminds us of the purpose of the ball. That is worked only upon faithfulness of the those who were using it to direct them on their journey. If they were not living as they should, they would be on their own and suffer hunger and contention as people always do who think they can do it on their own.
[I think we can compare the Book of Mormon to the ball or Liahona. It does give us revelation from time to time as needed if we live as we should. I know I have had suggestions given to me that had nothing to do with the text I was reading at the moment.]

Mosiah 1:18 And now, it came to pass that Mosiah went and did as his father had commanded him, and proclaimed unto all the people who were in the land of Zarahemla that thereby they might gather themselves together, to go up to the temple to hear the words which his father should speak unto them.

Mosiah did asked the people to gather around the Temple to hear king Benjamin.
[I doubt that this gathering of people could have happened in one day. They must have already been in town. It was probably one of the Jewish festivals that the law of Moses required of people to celebrate during the year, such as new years.]