Avraham administered an oath to Eliezer to ensure that he look for a bride for Yitzchak only from Avraham’s own family and not from the neighboring tribes. The Brisker Rav asks why an oath was necessary in this case. Tosfos (Kesubos 7b d”h she’ne’emar) writes that Eliezer served as a shliach, an agent, to effect the kiddushin of Yitzchak (see Maharil Diskin who discusses exactly how this shlichus worked, as under normal circumstances an eved may not serve as a shliach). If Reuvain appoints Shimon to serve as his agent to buy a red Toyota, but instead Shimon buys a green Ford, Reuvain doesn’t have to accept that green Ford – since Shimon violated the terms of his appointment, Shimon’s actions on Reuvain’s behalf are worthless. Since Eliezer was appointed only to take a bride from Avraham’s own family, kiddushin made with any other family should be automatically null and void even without an oath.
The Brisker Rav leaves this one unanswered, but I am wondering if there is a solution. The pesukim seem to suggest that Avraham was open to the possibility of marrying Yitzchak to someone else should Eliezer not find a willing bride from his own family. Perhaps Eliezer was appointed as an agent with a blank check – find a bride, ideally from family, but if not, another suitable candidate. If Reuvain appoints Shimon to buy a car and expresses a preference for a red Toyota, but is willing to accept a green Ford if there are no good deals on that red Toyota, Shimon has a lot more leeway. Perhaps an oath was necessary to cut down on Eliezer's leeway and guarantee he will make every effort to find a bride from Avraham's own family.
[Update: I should have thought about this more. The Brisker Rav probably also had in mind that Eliezer had a blank check shlichus. The question the Brisker Rav is driving at I think is why do things in that way -- why not appoint Eliezer a shliach to find a bride only from Avraham's family, which would eliminate the need for an oath.]
Tosfos writes that we learn from this incident that birchas eirusin may be recited when kiddushin are done by a shliach. Why would one think not? My first reaction is that Tosfos is getting involved in the classical question of who and whether a birchas hamitzvah can be recited where the ma’aseh mitzvah is done by one party but the kiyum mitzvah belongs to a different party. If birchas eirusin is a birchas ha’shevach and not a birchas hamitzvah, as some Rishonim indicate, then perhaps one might have thought there is no place for a birchas hashevach when the groom is not present to appreciate the new status of eirusin being created – again, this is basically the same idea of this case being special because the action is performed by one party but the kiyum, chalos, enjoyment, etc. coming to another.
My son did not like any of these type approaches. He reads Tosfos as telling us a chiddush in birchas eirusin in particular, not as a general rule that applies to all brachos or all other mitzvos. If that is the case, the question remains (which he does not yet have an answer for) as to why one might have thought birchas eirusin is unique.