The following only addresses the absence of his name in Parshas Chayei Sarah, but that is the bulk of cases in which his name is replaced with eved/ish.
R' Yaakov Kaminetzky (Emes L'Yaakov Bereishis 24:39) sees in this point an important motif underlying the story of Eliezer's search for a wife for Yitzchok. Eliezer had a daughter whom he wanted to marry off to Yitzchok, but this was not accepted by Avraham because Eliezer was a descendant of Canaan and was accursed, whereas Yitzchok was blessed (Rashi to Bereishis 24:39). This was reflected in his being an eved (servant), as this was the curse that Canaan was given (Bereishit 9:25). R' Yaakov explained that during the course of Eliezer's mission, he started out as "eved" (verses 2 - 17), but upon finding Rivka and completing his mission, he became "blessed" (as Lavan declares in verse 31), and therefore was transformed into an "ish" (verses 21 - 32). However, after having removed his ineligibility to marry his own daughter to Yitzchok, Eliezer tried to ruin the match by implying there was some problem with Yitzchok (verse 39verse 39), and at that point he reverts to being an eved (verses 52 - the end, with two exceptions that R' Yaakov explains). So the absense of his name is in order to refer to him by his status, which is important to the story line.
(I have my own explanation of this labeling pattern, but I'll defer to R' Yaakov for this forum.)
