My son was complaining to me that if I don't write anything on the parsha it will be 2 straight weeks with nothing...
Had Eliezer followed Avraham's instructions he would have travelled to Aram Naharayim, asked around to find Avraham's family, knocked on the door, and hoped to find a suitable match for Yitzchak there.
But that's not what he did. Aside from the whole test he setup to find a girl who excels at doing chessed (and if Avraham didn't tell him to do that, we should ask ourselves why Eliezer thought it necessary, but that's a different topic...), the Torah tells us that he davened. "Vayomar Hashem Elokei adoni Avraham hakei na lifanay hayom v'aseh chessed im adoni Avraham..." (24:12). According to Rashi, even the conclusion of Eliezer's description of the test he was setting up, "u'bah eidah ki asisa chessed im adoni," (24:14) is not a statement of fact. i.e. the test proves that Hashem did a chessed and he found the right girl, but rather is part and parcel of his tefilah, i.e. "u'bah eidah..." through this test I pray that I will know who the right girl is. It's not a tefilah + a test, but it's one big tefilah that things work out (Ramban and others disagree).
I think it's safe to assume that if we would say a little tehillim before embarking on an important project, the Avos and the people in their household would do no less. Had the Torah not told us that Elizer davened, would we have assumed otherwise? Of course not. So while the details of the test might be necessary and relevant to the story, the question begs itself: why does the Torah feel the need to stress Eliezer's tefilah here?
The Sefas Emes asks the question, but I want to offer a different answer than the one he gives.
There is an interesting machlokes in the meforshim how to interpret the tefilah of Eliezer. Most (Targum Yonasan, Ibn Ezra, Ramban, Kli Yakar) interpret "hakrei na lifanei" to mean that Hashem should cause the right girl to appear before Eliezer. However, Alshich and more clearly the Netziv read it differently. Before Eliezer left Avraham's home, Avraham gave him a bracha: "Hu [Hashem] yishlach malacho lifanecha..." Hashem should send his angel along with him to help (24:4). Now Eliezer gets to Aram Naharayim and he davens to Hashem, "hakrei na lifanei," you, Hashem, please appear before me. It's not enough for me to have the help of a malach to pull this off -- I need you here with me.
I think the Chazon Ish is quoted as saying that the last remnant of open hashgacha that we have left is in inyanei shidduchim. It's not the shadchan, it's not even malachim, but it's Hashem himself who makes a match happen, even sometimes in the most unlikely situations.
That's perhaps why the Torah makes a point of including the tefilah of Eliezer. Of course everything the Avos did was accompanied by tefilah. But when it comes to the parsha of marriage, tefilah is not just a nice thing to do to accompany the mitzvah, but it is part and parcel of the mitzvah itself. Since direct intervention of the yad Hashem is necessary, tefilah, dveikus with Hashem, asking Hashem for that direct involvement, is a must.