I've heard that point used to show how devoted Eliezer was. He was this big hotshot, making a match for the young master. But he felt so unimportant, that he never even mentions his name. When talking to Rivkah's family, he introduces himself by saying, "I am a servant of Abraham."
In yeshiva they would tell us to learn from Eliezer and not take ourselves so seriously.
Yosef Linkstein Twin Cities, MN
November 21, 2008
I don't believe the parsha mentions Eliezer specifically as the servant.