Avraham made Eliezer promise that he would only find a wife for Yitzchak from his family.
It seems to follow then, that as soon as Eliezer would arrive in Aram Naharaim he'd go straight to Avraham's family. After all, it was only a small family with very few (a couple of?) options to choose from (Rivkah,Ma-acha,?) - [Edit: being that 'family' here seems to refer only to descendants of Nachor (see Ramban on Bereshis 23:24- who suggests that for this reason Nachor's lineage from both wives are mentioned in the end of Parshas Vayeira)]
So why did Eliezer have to make the whole sign/prayer?
On the contrary things were more probable to go wrong this way. (See Gemara Taanit 4a)
[PS: This question is related, but certainly different.]