Why did Samuel think Israel should have a king?

Answer 1

1) He served as Judge (1 Samuel 7:15) and leader. Each year, he traveled in order to teach and serve as Judge (1 Samuel 7:16) in the various places.
2) He guided the people when they requested that a king be apointed (1 Samuel ch.8); and he apointed Saul and King David at the word of God (1 Samuel ch.9-10 and ch.16).
3) He wrote the Books of Ruth and Judges (Talmud, Bava Batra 14b) and the Book of Samuel (until the verse concerning his death; it was completed by another prophet. Talmud, ibid) .
4) He led the people in a great movement of repentance (1 Samuel ch.7).
5) Until his time, prophecy had been relatively rare (1 Samuel 3:1). Starting with his leadership, the phenomenon of the "sons (disciples) of the Prophets" (2 Kings 2:3) became a regular institution, and very many people reached the various levels of prophecy (Talmud, Megillah 14a) (though only fifty-five prophets were commanded to record words for posterity. Ibid).

Answer 2

The question is incorrect. The Prophet Samuel was actually opposed to the idea that Israel should have a king, believing that God alone was needed for Israel to be governed. However, God convinces Samuel to appoint a king for the people since they want one and Samuel does this by appointing Saul and later David.

Answer:

The Israelites at that time were coming to the end of an era which was unique in world history. They had undertaken a grand experiment: whether a nation could govern themselves for centuries without a king or organized government.

Instead, there were the officers of tens, fifties, hundreds and thousands (Exodus ch.18), and the court of Elders, who were Torah-scholars that provided Torah-rulings and guidance. Each of the leading Judges (Gideon, Deborah etc.) was a private citizen (not a head of government) who led the nation only during a brief episode of battle.

During that era (of the Judges, about 350 years), when someone raised the possibility of having a king, the answer was: God will rule over you, not a king (Judges 8:23). The events of the Exodus and the Giving of the Torah were so fresh in the nation's memory that they didn't need a king; God was their King. (The missteps which did take place in that era, is a subject beyond the scope of the present question.)

In Samuel's old age, nearly four centuries after the Exodus, the people (including their Torah-scholars) felt that the time had come to take a regular government. The Torah itself permitted this (Deuteronomy ch.17); and they saw that Samuel's sons didn't seem to have reached his spiritual level (1 Samuel 8:2-3).

Their usage of the expression "like the other nations around us" was not a problem, since they were deliberately quoting the Torah (Deuteronomy 17:14).

What then was their mistake (why did Samuel criticise them)?

According to the Malbim commentary (on 1 Samuel), they should have waited at least until Samuel was too old to function.

According to the K'li Yakar commentary (on Deuteronomy), their precise choice of language ("for us," instead of the "over us" which the Torah had said), hinted that they wanted a king who might be affected by public pressure (which later happened with Rehoboam). See Talmud, Yoma 22b.

According to Samuel himself (1 Samuel 8:10-18), they were taking a regrettable risk because later kings might be overbearing.