Why did Moshe build a Mizbe'ach?
Ha'amek Davar: To thank Hashem for the good news that He would erase the name of Amalek. 1
Gur Aryeh: See below, 17:15:2.1:1 .
Presumably, he brought Korbanos on it, as Targum Onkelos intimates.
What did Moshe call 'Hashem Nisi'?
Rashi and Ramban: "Anyone who mentions the name of the Mizbe'ach, will recall the miracle that Hashem did for us." Compare to Bereishis 33:20:1:1 .
Seforno: He called the name of Hashem in prayer, 1 'Hashem Will Be My Champion' against all those who rise against me. He called the Mizbe'ach, 'Hashem is my flag-pole' 2 - because Hashem's staff served as a flag-pole on the hill (to rally the army to fight against Amalek). 3
Rashbam: He was calling the Mizbe'ach by the name 'Hashem Nisi,' because Hashem held His staff aloft on the hill, 4 which served as a flag for the army of Yisrael to come and fight Amalek. 5
Targum Yonasan: He called the name of the Mizbe'ach 'The Word of Hashem, this is my miracle; because the miracle that Hashem performed when His Shechinah passed here was for my sake.' 6
Seforno: As in "Karasi Shimcha Hashem" (Eichah 3:55).
Seforno: As in the Pasuk, "Nasata li'Yere'eicha Neis l'Hisnoseis" (Tehilim 60:6).
See also Ba'al ha'Turim.
And in the same way, Hashem will raise His staff on the mountains, to call the armies to fight against Amalek (in the time of Mashi'ach) as the Pasuk continues, "Ki Machoh Emcheh ...."
The Rashbam's explanation is similar to that of the Seforno - See answer #2 and note 2 - only, where the Seforno interprets it as a Tefilah, the Rashbam interprets it as a statement.
Since Yisrael were not worthy of miracles at that stage (See Na'ar Yonasan).
QUESTIONS ON RASHI
Rashi writes: "'Hashem Nisi' - Anyone who mentions the name of the Mizbe'ach, will recall the miracle that Hashem did for us." If so, why specifically a Mizbe'ach?
Gur Aryeh: It is as a result of our Avodah before Hashem, that He saves us from our enemies and performs miracles for us; such that each person can say, "Hashem is my miracle." A Mizbe'ach is the place of Avodah.
