When reading the first few פסוקים of פרשת יתרו, one should ask why the names of Moshe's children are mentioned here. Ramban explains that it is because Eliezer's was not mentioned earlier since it would not fit into any earlier narrative. It is appropriate at this point, however, to introduce Eliezer's birth, because now he can truly say אלקי אבי בעזרי (for saving Moshe from פרעה, splitting the Yam Suf, and drowning the Egyptians), and therefore Eliezer's name now fits into the narrative.
One, however, could ask about the Ramban that the thanks Moshe offers is only for saving him from Paroh but not for splitting the ים סוף: This would have seemed to fit in פרשת שמות for Moshe to say אלקי אבי בעזרי ויצילני מחרב פרעה. It is possible to suggest that Eliezer's name is not mentioned in פרשת שמות because of the delay of his ברית מילה (see Rashi to שמות ד: כ"ג-כ"ו). When a male has his Brit, he receives a name. Hence, the delay in the Torah's recording of Eliezer's birth may be a reflection in the delay of his Brit.
The link of ברית מילה and receiving a name is a recurrent theme in Jewish practice and the חומש. We find the link by אברהם, and also יתרו changed his name when he converted (see Ramban to במדבר י:כט) Rambam notes that the custom of גרים is to give themselves a new name that reflects their new identity. Conversely, the name of עפרון החתי was reduced by a letter after he showed that he not only was an עובד עבודה זרה and therefore did not acknowledge Hashem, but he also disregarded Hashem's בין אדם לחבירו laws by lying to Avraham when he first said Avraham could have Me'arat Hamachpeilah for free and subsequently charged him an exorbitant 004 Shekalim (see Rashi to בראשית כג:טז).
We see that one's name is enhanced by the mitzvot he does and diminished by lack of mitzvot or sins. This is probably what Shlomo HaMelech meant when he wrote in קהלת -טוב שם טוב משמן טוב- "a good name is more valuable then precious oil."