Socio-cultural background, casting lots: While modern people might be familiar with casting lots to create a pure chance selection, it would be a disservice to these brothers to assume that the casting of lots was done to create a random assignment. Casting lots in the ancient world assumed that the randomness inherent in the lots opened the door for God to place His hand in the outcome. The casting of lots was used as a means of discovering the will of the Lord.
Laman, Lemuel, Nephi, and Sam had been sent on a mission by their father. The mission had been commanded of Lehi in a prophetic vision. Regardless of the feelings that Laman and Lemuel had about their state of affairs in the wilderness, they were familiar with religion, and knew that this was a religiously directed mission. In a more modern story, we might see the four brothers kneeling in prayer to determine who should go to Laban. The casting of lots was essentially the same for them.
Nibley has a fairly long description of ancient divination practices in his Since Cumorah (Deseret Book 1970, pp 287-292). To distill that information, the divination was frequently one where rods would contain the words "My Lord hath commanded me" and some would have "My Lord hath forbidden me". Action would be taken based on the message which appeared on the chosen lot (see Nibley 1970 p. 288). Thus the casting of lots was an action equivalent to asking the direction of the Lord, which would be consistent with the religious purpose of this quest into Jerusalem.
Daniel H. Ludlow also recognizes the practice of casting lots in the Old Testament, and cites examples of where lots were used in the scriptures: Leviticus 16:8, 1 Samuel 14:42, 1 Chronicles 26:13, Psalms 22:18, Isaiah 34:17, Joel 3:3, Obadiah 11, Jonah 1:7, Nahum 3:10, Matthew 27:35, Mark 15:24, John 19:24, Acts 1:26 (Ludlow, A Companion to Your Study of the Book of Mormon . Deseret Book 1976, p. 95).
Scriptural analysis: The use of the lots to determine who should go up to Laban now raises another question. If the Lord did have a hand in the selection, why was Laman chosen instead of Nephi? Clearly the Lord knew that Nephi would be the successful one. Why was Laman allowed to fail?
There are two issues here. The first is why might Laman be allowed to fail, and the second was why Laman was chosen instead of Nephi. Laman was allowed to fail for the same reason that all of us are allowed to fail at various tasks in this life. Our Agency places us in a position where we must individually do or fail at any number of tasks. Opportunities are placed before us, and we will succeed or not depending upon our developed abilities to handle each situation. There is therefore no surprise that the Lord would allow Laman to fail. In such an important mission, it is likely that the Lord would not let the mission fail, only one of the men who attempted to carry out the mission. Laman's eventual failure was personal. The goal of the mission was eventually accomplished.
The next question is why the Lord would put Laman in a position where the Lord knew he would fail. The incident of the retrieval of the brass plates comes directly after Nephi's epiphany wherein the Lord told Nephi that he would ascend over his brothers. While that had occurred in other occasions (Jacob over Esau, Joseph over all his brethren) it was remarkable in its violation of cultural rules. The eldest brother was the one who should have that role.
The incident with the brass plates is a transitional moment where the change in the leadership hierarchies is being established. The Lord uses this incident to provide justification for the ascendency of Nephi as a leader, a process which continues on their journey through the desert. Laman's attempt and failure was both a recognition of his birthright, and the marking of the passage of the birthright from the inheritor to the more righteous anointed.
Historical analysis: As Nephi begins this task, he is a younger brother with no social claim to preeminence over his brothers. While we are not given the ages of the siblings we can make some guess as to Nephi's age at this point. From the departure from Jerusalem to the end of Nephi's life is approximately 55 years (see Jacob 1:1-12). Since one of the tasks before finally leaving forever would be to secure wives for the brothers, we might make the assumption that Nephi would have been of marriageable age, which at that time might have been in the early or mid teens. That would give Nephi a lifetime age of around 70 years, which fits within other ages in the Book of Mormon. Thus at this time, we might expect Nephi to be in his teens (perhaps mid teens?) with Laman, Lemuel, and Sam one to two years older each.
Even though Nephi is physically mature, he is not yet of an age where he can command respect. As this episode begins, he is very much a young man ready to be forged into the man he will become.