ויהי הוא טרם כלה לדבר, "and it was before he had finished speaking, etc.⁠" We need to understand the need for the apparently extraneous word הוא in this verse. All the Torah had to write to tell us that Rivkah appeared before Eliezer had completed his prayer was ויהי ככלותו לדברו, "as soon as he had finished speaking, etc.⁠" Actually, the word הוא in this verse is not a personal pronoun, i.e. "he,⁠" but a name of G-d. (compare Genesis 19,33 where we encountered this word in that context) We have numerous examples of the word הוא being a reference to a name of G-d, such as Psalms 100, 3 הוא עשנו ולו אנחנו, "He has made us and we belong to Him;⁠" Numbers 18,23 ועבד הלוי הוא, "and the Levite will serve G-d, etc.;⁠" the word הוא in that verse means that Eliezer enjoyed G-d's special assistance in his mission by means of the angel concerning whom Avraham had prayed to G-d that He should make Eliezer's mission successful (verse 7). The angel, was at hand already before Eliezer had completed praying and he arranged for Rivkah to be the maiden whom Eliezer would address first with his request for water. This is why the Torah wrote: והנה רבקה יוצאת, "and here Rivkah was already in the process of coming out, etc.⁠" The word והנה means that what follows was the result of a special arrangement, invitation. If the servant immediately ran towards her (verse 17) it was because he had noticed the water of the well rising to meet Rivkah on her descent. This is what our sages in Bereshit Rabbah 60,5 have derived from the wording in our verses here.
At that moment Eliezer already understood what Avraham had meant when he had said to him that G-d would send His angel ahead. This is also what he related once he was in Bethuel's house when he told the family that Avraham had assured him already before he set out on his journey that G-d would send His angel ahead and that he would succeed (verse 40). In view of this it is easy to understand that from the moment that Eliezer had become aware that the angel was supporting him the Torah refers to him no longer as עבד, servant, but as איש, a man of stature. Up until then we read about וישם העבד, "the servant placed;⁠" ויקח העבד, "the servant took;⁠" וירץ העבד, "the servant ran.⁠" From this point on the Torah keeps referring to Eliezer as האיש until he leaves the house of Bethuel with Rivkah. The expression האיש is applied to the angel Gavriel in Daniel 9,21. It is also applied to the angel who told Joseph that his brothers had moved away from Shechem, (Genesis 37,17) and that they had been overheard as planning to move to Dothan. In other words, as of the time the angel joined Eliezer in his quest, he himself assumed the role of the angel in an executive manner.
The words והאיש משתאה לה (verse 21) which appear to be somewhat problematical if we assume Eliezer and the angel to have become as one, must be understood as "he was waiting for her;⁠" he did so as he had become aware that his mission was about to be crowned with success. Having done so, the Torah from now on refers to him constantly as האיש, such as when we read (verse 22) ויקח האש, or (verse 26) ויקוד האיש, or (verse 30) כה דבר אלי האיש, etc. When, a little while later, Lavan and Bethuel gave their consent for Eliezer to take Rivkah (verse 51) and they said הנה רבקה לפניך קח ולך, "here Rivkah is in your presence, take her and go,⁠" at which point Eliezer's mission had been completed, the Torah reverts to describe Eliezer as עבד instead of as האיש. In verse 53 when Eliezer is reported as sharing out gifts, as well as in every subsequent move of his, the Torah speaks of העבד, "the servant,⁠" seeing that the angel's assistance was no longer required for that part of his activity. Avraham's prayer which invoked the assistance of an angel had been answered in full.
If you were to ask that when Lavan and his mother asked Rivkah if she were willing to go עם האיש הזה, "with this man,⁠" and this occurred only in verse 58, after verse 51 which we described as the point at which Eliezer's mission had basically been completed, the answer is simple. The words reported in verse 58 were not spoken by the Torah but by Lavan and his mother. They certainly were not bound by the considerations we described before. How could they be expected to refer to someone as "this slave,⁠" when the person in question was distributing lavish gifts to every member of their household, not to mention to Rivkah herself? Would it not have been a gross insult to their sister to ask her is she were willing to leave her parental home in order to travel to a distant land "with this slave?⁠"
Concerning verse 61 in which both Rivkah and her maids are reported as traveling אחרי האיש, which appears to contradict the explanation I have offered that Eliezer had reverted to being an עבד the moment consent had been obtained for her to become Yitzchak's bride, this wording was chosen by the Torah in honour of Rivkah. It would not have looked respectful if she had been described as riding "behind the slave.⁠"
Another approach to the words והאיש משתאה לה. It is an allusion to the angel who had tarried and waited at the well until the arrival there of Rivkah. From that moment on Eliezer took over the task of the angel; hence he is referred to as האיש from that moment on. Support for this theory is found in the text near the end of the episode where the Torah said (verse 61) "they walked behind האיש, "and the עבד took Rivkah and he went.⁠" This verse makes it crystal clear that the subjects איש and עבד respectively could not have been one and the same.