Eliezer introduces himself simply as Avraham's servant when he reaches the family of Rivka (Bereishit 24:1-67), and already earlier in Avraham's own words he is described as 'ben meshek beiti,' the one in charge of the household and presumed heir before Yitzchak was born (Bereishit 15:2-3). The Gemara reads this verse midrashically, deriving from the name 'Dammesek Eliezer' that he was an elder who sat in the academy and had mastered his master's Torah, drawing and giving others to drink from it (Yoma 28b:8), a reading that Rashi brings as well, citing the same notarikon of doleh u'mashkeh (Rashi on Bereishit 15:2).
Chazal single out Eliezer for extraordinary merits: he is counted among three for whom the land miraculously contracted so that his journey to the well was completed in a single day (Sanhedrin 95a:16), and he is described as equivalent in strength to Avraham's three hundred eighteen men, since the gematria of his own name equals that number (Nedarim 32a:17).
The midrashic tradition also frames Eliezer as a model of wise, faithful service: he is called an 'eved maskil' who feared Hashem and listened to his master's voice even on a difficult mission (Yalkut Shimoni on Torah 107:3), and he is grouped among those whose words were answered before they finished speaking, alongside Moshe and Shlomo (Yalkut Shimoni on Torah 108). Yet Chazal also count him among three who prayed in an improper manner, even though his request at the well happened to be answered favorably (Taanit 4a:4-5).
Some sources trace Eliezer's background to the house of Nimrod: one midrash has Nimrod himself giving Avraham his son Eliezer as a servant forever (Midrash HaGadol), while the Chizkuni and Midrash Aggadah connect the name Dammesek Eliezer to Nimrod's grandson or to his pursuit of the kings until Damascus (Chizkuni, Bereishit 15:2, Midrash Aggadah, Bereishit 15:2).
Chassidic sources develop Eliezer as a figure of transformation through service: the Sefat Emet explains that although Eliezer descended from Canaan, under the curse, his faithful service to Avraham lifted him from the category of 'cursed' into that of 'blessed,' making him the paradigm of the 'eved maskil' whose rectification comes through service rooted in faith (Sefat Emet, Bereishit, Chayei Sara 27:4, Sefat Emet, Bereishit, Chayei Sara 23:5), and elsewhere compares the servant's role to that of the body serving the soul (Sefat Emet, Bereishit, Chayei Sara 30:4). The Zohar, as explained in the Sulam, likewise identifies the 'servant of his master' sent to fulfill his master's will as Eliezer being sent to Charan to arrange Yitzchak's marriage, bringing honor to Avraham (Sulam on Zohar, Vayera 143), and the Shaarei Orah draws a parallel between Eliezer's faithful service enriching Avraham's house and the cosmic service that draws down blessing (Shaarei Orah Shaar HaChanukah דף יט,ב).
Other passages depict Eliezer's lasting closeness to Avraham even after death: the Gemara in Bava Batra recounts Rabbi Bena'ah finding Eliezer standing at the entrance to the Cave of Machpelah and conversing with him about Avraham's position within (Bava Batra 58a:3-4), a scene Asara Maamarot elaborates upon to show Eliezer continuing to serve Avraham and Sarah in that realm (Asara Maamarot, Em Kol Chai 3:30). The Ramban also discusses the episode of Eliezer loosening the camels' muzzles, weighing the comparison in the midrash between Avraham's camels and Rabbi Pinchas ben Yair's donkey as a measure of the household's piety (Ramban on Bereishit 24:32).
On the matter of Eliezer's conduct at Rivka's family's table, one source explains that he delayed eating until he had delivered Avraham's message, reasoning that prioritizing his master's affairs over his own needs would protect him from the ill effects of eating at a begrudging host's table, and it concludes that this merit indeed shielded him (Valle on Bereishis 24:34).
Further sources treat Eliezer as representative of all of Avraham's servants, reflecting Avraham's own greatness (Chiddushei Aggadot, Nedarim 32), as the agent who merited witnessing many miracles in bringing forth Rivka, the precious 'pearl,' from a lowly family (Valle on Bereishis 22), and as the servant whom Avraham made swear by the mitzvah of milah precisely because that mitzvah was so dear to him (Beit Yaakov, Beshalach 31). Rashi on Sanhedrin likewise simply identifies the Eliezer mentioned there as Avraham's servant (Rashi on Sanhedrin 108b:19:1), and later sources continue to reference his identity as Avraham's servant directly (Hegyonot El Ami, Volume II, Chayei Sara 22:1).
The Yalkut Reuveni traces a mystical lineage of gilgulim beginning with Eliezer's soul, said to have passed through the dog that lay upon Avraham's grave and onward through later figures such as Benayahu ben Yehoyada and Zecharya (Yalkut Reuveni).
A separate halachic question arises from Eliezer's status: since the Gemara in Kiddushin discusses the honor due to Rabban Gamliel's household servants in a related context of service and deference (Kiddushin 32b:4-5), the Klausenberger Rebbe directly addresses how Eliezer could have functioned as Avraham's agent in matters like betrothal if he was in fact a slave, concluding that he must have been freed, or alternatively that the prohibition on keeping a servant in permanent bondage did not apply because his service was bound up with performing a mitzvah (the Klausenberger Rebbe (Divrei Yatziv YD 185)).