[Yitro brought Tziporah,] and her two sons; of which one's name was Gershom, because [Moshe] said "I have been a foreigner in an alien country"; and the other's name was Eliezer, because "my father's God was in my aid, and saved me from Pharaoh's sword". (18:3-4)
The commentators rightly point out that while Gershom's birth was mentioned when it occurred, this is the first mention of Eliezer. And once the Torah decided to mention Eliezer (and explain his name), it mentioned Gershom too (and explained his name) for completeness, even though this repeats information we already know from Shemot 2:22.
This explanation of the Gershom-related repetition seems technically reasonable, but I don't find it intuitively satisfying. Or, to put things more clearly, I think it explains the causes of the repetition, but does not attempt to explain the repetition's effects.
As far as the effects go, I think the names must be reexamined now that the Israelites have left Egypt for good. Eliezer was named to commemorate God's saving Moshe from Pharoah after Moshe killed the Egyptian taskmaster. Now, when Moshe is leaving Egypt after the plagues, the name takes on a whole new significance. In the plagues, Moshe had a more direct and violent confrontation with Pharaoh than after killing the Egyptian in his youth. And the degree of God's assistance in the plagues was much, much greater. In retrospect, Eliezer was better named for the events following his birth than for the events before them.
The same is true, with a twist, of Gershom. Moshe had originally meant that he was a stranger in the strange land of Midyan. As he understood it, Egypt was the native country he was forced to leave. But upon leaving slavery, the name Gershom resonates with a different set of historical associations. In truth, Egypt itself was the strange land, which Moshe and the Israelites were leaving after living there as strangers for centuries.
In each case, Moshe named his sons for personal reasons. But when the Israelites leave Egypt, the names come to represent the entire nation's experience. This indicates that Moshe has been transformed from a solitary individual to a leader wholly immersed in the concerns of his people, and points us to a future in which the entire nation's relationship with God is as intimate as Moshe's has been.
And you shall provide from all the people, men of valor, God-fearers, men of truth, who hate unjust gain; and you shall place them as leaders of thousands, leaders of hundreds, leaders of fifties, and leaders of tens. (18:21)
Rashi: men of valor - wealthy men, who have no need for flattery or playing favorites.
men of truth - these are people of whom it's known that it's worth relying on their words; thus their words will be accepted.
who hate unjust gain - who hate [i.e. don't bother pursuing] their money in court, as the gemara says: "Any judge whom money is taken from in court, is not a judge".
The verse seems to be simply asking for people who are honest. But in these three comments, Rashi takes it in a different direction. For him, the quality being sought is public knowledge, not simply the existence, of a person's honesty. A hidden tzaddik would not qualify to be a judge. Rashi is pointing out that a justice system cannot be effective unless the public trusts it and is willing to rely on it.
And Elokim spoke all these words, saying: "I am Hashem your God, who took you out of Egypt..." (20:1-2)
It's beyond the scope of a single blog post to explain every instance in the Torah in which the name "Elokim" is used and not "Hashem" or vice versa. But in this particular case, the reason for the switch between names seems clear.
Compare this verse to the beginning of parshat Vaera: "And Elokim spoke to Moshe, and said to him: 'I am Hashem. I revealed myself to Avraham, Yitzchak and Yaakov as El Shaddai..." (6:2-3)
In both cases, God makes a declaration of immense thematic significance, consisting of a declaration about God followed by commandment(s). In both cases the first words of the declaration are "I am Hashem", and it's the first time that God has used that name to address Moshe/the Jewish people. (At the burning bush, God did not use any name until Moshe asked for it.) And in both cases, the declaration is introduced using the name "Elokim".
It therefore seems that Moshe's experience in parshat Vaera is his personal equivalent of what would took place for the whole people at Sinai: the forging of a new, permanent covenant using the name "Hashem" and all that is associated with it. Before this covenant, the relationship between God and Moshe/Israel was at a level symbolized by the name "Elokim". (To use R' Soloveitchik's terminology, it was a "covenant of fate" and not yet a "covenant of destiny".) With the forging of the covenant, the relationship took a sudden quantum leap forward. This qualitative change is indicated by a change in the name of God used - from "Elokim" to "Hashem", both in parshat Vaera and at the giving of the Torah.