To what extent does mainstream Christian doctrine teach that knowing a spirit's name gives you power over it?

I've seen this question asked often in regards to exorcism of demons, so allow me to offer an analogy to explain things as I understand them. (Disclaimer: I'm not a theologian.)

A robber (demon) breaks into a store (possessed individual) which triggers an alarm that is subsequently detected by a security guard (exorcist.) Initially, the security guard isn't certain whether or not the alarm was simply a false alarm, so he performs his routine to search for evidence of a break-in (possession.) From the robber's perspective, the ideal outcome is that his presence is never detected at all. Obviously, he can't be caught and stopped if he evades detection. If the security guard is able to track him down, expose, and corner him, his only remaining task is to catch (exorcise) the robber. The robber/demon alerting you to his presence is counter-productive to his ultimate goal. So it stands to reason that he will avoid this whenever possible.

Thus, in this analogy, the demon telling you his name is equivalent to the robber jumping around, flailing his arms, and yelling, "Here I am! Here I am!" This gives the security guard a greater edge over performing his duties, while simultaneously making the robber's job that much more difficult.

The above is informed by a concept harvested from a book called Mechanics of Demonology, written by exorcist, pastor, and expert Christian Theologian G.P. Haggart.

Taking a slightly different tack (and not wanting to detract at all from fredsbend's excellent and informative answer), the short answer is a definite No.

Any authority over evil spirits that a believer may exercise does not derive from knowing their names, but comes by way of delegation from Christ (cf. Matthew 28:19-20;10:1; Luke 10:17-20; Mark 16:17).

The Legion case was unusual. It seems likely (to myself, at least) that Jesus did cast a demon out with his initial command, but thousands remained. In order to expedite the process and to set an example for his disciples when they encountered a similarly unusual situation, Jesus asked for a name in just this particular instance. There are no other recorded instances of him asking for a demon's name. The fact that the demons instantly answered (they had no option to evade giving their name/description but to obey the Son of Man), undermines completely the sense of the teaching in question.

The teaching I have seen regarding exorcism as to why Christians actually do sometimes employ the practice of asking for demon's names, is to discern root causes of demonization with the intention of removing potential footholds (for re-demonization - cf. Matthew 12:43-45) post exorcism (cf. Deliver us from evil, Don Basham).

I’ll try to answer the question from a Catholic perspective and, additionally, provide some supplementary information about what has been mentioned in the comments.

The general answer to the question is “no”. First, God is Almighty, both before and after His revelation to us. So you can’t exercise power over Him. We can do wonders in Jesus’ name, but only in agreement with His will.

Second, in the case of angels, knowing their names doesn’t make us superior to them. For example: we can’t order anything to the archangel Gabriel just because we know his name. It’s useless, and it shows an absolute lack of love as well.

But,

in the case of demons, and only if you are an exorcist dealing with a possession, finding out the name of the possessor(s) certainly can give you an edge. Fallen angels’ behavior is quite sensitive to names, both those of themselves, and those of the ones living in Heaven (God, His angels and His saints), proving the relevance of names.

The following quotes belong to the book "An exorcist tells his story" by Fr. Gabriele Amorth (Ignatius Press 1999, twelfth edition), who is an Italian priest and exorcist. He's also one of the founders of the International Association of Exorcists.

(page 79)

The demons are very wary of talking; they must be forced to speak, and do so only in the most severe of cases, those of true and complete possession. When demons are voluntarily chatty, it is a trick to distract the exorcist from his concentration and to avoid answering useful questions when they are interrogated. In our questions we must hold to the following rules of the Ritual: Never ask useless questions or out of curiosity. We must ask for the name, whether there are other demons and how many, when and how the evil one entered that particular body, and when he will leave.

(pages 93-4)

The first thing that must be asked is the name. For the demon, who is so reluctant to reveal himself, revealing himself is a defeat; even when he has revealed his name, he is always reluctant to repeat it, even during following exorcisms. Then we command the evil one to tell how many demons are present in a particular body. There can be many or few, but there is always one chief, and he is always the first to be named. When the demon has a biblical name or one given in tradition (for example, Satan, Beelzebub, Lucifer, Zebulun, Meridian, Asmodeus), we are dealing with “heavyweights”, tougher to defeat. [...] The strength of possession is manifested also from the reaction of the demon to holy names. Generally the evil one does not and cannot say those names; he substitutes expressions such as “he” (referring to God or Jesus) or “she” (referring to our Lady). Other times he says, “your Boss” or “your Lady”, to indicate Jesus or Mary. If the possession is very strong and the demon is high ranking [...] then it is possible for him to say the name of God and Mary, always followed by horrible blasphemies.

(page 116)

Even the names of the demons, as in the case of angels, tell us their function. The most important demons have biblical names or names transmitted to us by tradition: Satan or Beelzebub, Lucifer, Asmodeus, Meridian, Zebulun. Other names more clearly tell us the purpose of their actions -Destruction, Perdition, Ruin- or they indicate individual evils -Insomnia, Terror, Discord, Envy, Jealousy, Sloth.