I hope that someone is still monitoring this thread. I, too, have been trying to get a better understanding of ἐξουσία.
I was unable to find ἐξουσία in Le Chantraine's le grand Dictionnaire étymologique de la langue grecque, nor could I identify any Greek words in the online edition of von Hjalmar Frisk's Griechisches etymologisches Wörterbuch, but Strong, Thayer, and Liddell and Scott all give the same explanation as billberg23: ἐξουσία=ἔξεστι (ἐκ + εἰμί).
Although I was unable to find any source supporting any other derivation, that one strikes me as odd for several reasons: (1) I have been unable to find any explanation for the claim that ἐξουσία is ἐκ + εἰμί. (2) I uncertain why εἰμί should become -ουσία; are the noun, ἐξουσία, and the related verb, ἐξουσιάζω, formed from the stem of the present participle? (3) Another word, ἔξειμι, is exactly what I would expect from ἐκ + εἰμί, but it is not used with this meaning. (4) It seems unusual that, if ἕξεστι is an impersonal form from ἔξειμι, it should have a completely completely different meaning than ἔξειμι (I go out). (5) ἐξουσία looks like it "should" be ἐκ + οὐσία. (6) An ἐκ + οὐσία derivation seems to fit the use of ἐξουσία better than ἐκ + εἰμί. (7) It seems to me that ἐκ + οὐσία helps to explain the examples where ἐξουσία is used in place of ἕξεστι, at least in the New Testament.
It appears that the central sense of οὐσία may be "property," or "that which pertains to or belongs to a person." If that is the case, what would naturally come "out of," or "be derived from," such "property" (using that word as a short-hand for the broader concept)? Would it not be rights, authority, or jurisdiction over that "property?" Importantly, the authority would not be arbitrary or based merely upon might, but upon the relationship to that "property." If so, might not one then be justified in interpreting ἐξουσία as approximately equivalent to "ownership," "rights over," or "mastery" (in the sense of the relationship between a "master" and the "servant" who is the "property" of that "master")? This interpretation also fits with the related use of ἐξουσία, given by Liddell and Scott as "abundance of means, resources"--in other words, ability or power arising from that which one has or to which one has access (ἐκ + οὐσία?)--but it is hard for me to reconcile that meaning with ἐκ + εἰμί, unless that meaning either was a misunderstanding of ἐξουσία, based upon a folk etymology of ἐκ + οὐσία, or else ἐξουσία (ἐκ + οὐσία) existed as a homonym with ἐξουσία (ἐκ + εἰμί).
In Matthew 8:9 and Luke 7:8, the centurion explains his own ἐξουσία, by placing himself "under" the same hierarchy that encompasses his soldiers and his slave; this would make perfect sense, if he and his soldiers were "the property of," or "that which belonged or pertained to" (οὐσία) the emperor, just as the centurion's beloved slave was his "property" (οὐσία), and ἐξουσία derived from (was ἐκ from that οὐσία), but his explanation would seem much weaker, if ἐξουσία is ἐκ + εἰμί.
If ἐξουσία is interpreted as ἐκ + οὐσία, Romans 13:1 would be, in essence, based upon the principle endorsed by Jesus, as recorded in Matthew 22:19-21, Mark 12:16-17, and Luke 20:24-25. Indeed, in Romans 13:7, Paul concludes his argument with an apparent allusion to that very teaching by Jesus: ἀπόδοτε πᾶσιν τὰς ὀφειλάς, τῷ τὸν φόρον τὸν φόρον, τῷ τὸ τέλος τὸ τέλος, τῷ τὸν φόβον τὸν φόβον, τῷ τὴν τιμὴν τὴν τιμήν. In contrast, this allusion loses much of its impact, if ἐξουσία is ἐκ + εἰμί: what point is there in an allusion to authority based upon "ownership" or "property rights," if the ἐξουσία of Romans 13:1 is based upon nothing more than mere hierarchy (as Matthew 8:9 and Luke 7:8 would imply, if ἐξουσία is ἐκ + εἰμί)?
In 1 Corinthians 11:3-16, the argument for a woman having a "casting around" the head (head-wrap), when she prayed or prophesied, was premised upon the headship of man, as based upon the woman having been "built out of" man, rather than the man "built out of" woman. This argument is taken directly from Genesis 2:23, right down to the "through" or "on behalf of" relationship. The obvious allusion to woman being "bone of my bone, and flesh of my flesh" (Genesis 2:23) implies that, since the woman is part of man's body, the woman "pertains to or belongs to" man in the same way that man's other body parts do (Paul seems to have interpreted that very concretely: οὕτως ὀφείλουσιν καὶ οἱ ἄνδρες ἀγαπᾶν τὰς ἑαυτῶν γυναῖκας ὡς τὰ ἑαυτῶν σώματα. ὁ ἀγαπῶν τὴν ἑαυτοῦ γυναῖκα ἑαυτὸν ἀγαπᾷ. οὐδεὶς γάρ ποτε τὴν ἑαυτοῦ σάρκα ἐμίσησεν ἀλλὰ ἐκτρέφει καὶ θάλπει αὐτήν Ephesians 5:28-29 and ὑμεῖς οἱ καθ’ ἕνα, ἕκαστος τὴν ἑαυτοῦ γυναῖκα οὕτως ἀγαπάτω ὡς ἑαυτόν Ephesians 5:33. As in Genesis 2:24, Paul also understood marriage as restoring the parts separated at creation into one body: ἄνθρωπος τὸν πατέρα καὶ τὴν μητέρα καὶ προσκολληθήσεται πρὸς τὴν γυναῖκα αὐτοῦ καὶ ἔσονται οἱ δύο εἰς σάρκα μίαν Ephesians 5:31.). Somehow, the creative relationship necessitates a wrap of the woman's head, while she is praying or prophesying, and the wrap serves as ἐξουσία "upon her head on account of the messengers." The connections between praying or prophesying, a head-wrap, ἐξουσία, and the messengers may not be clear to us, but, whatever the connection, it would make far more sense for Paul to have based his argument upon an ἐξουσία derived from woman's creation out of man's body parts, if ἐξουσία is ἐκ + οὐσία, than it would if it is ἐκ + εἰμί. While the underlying symbolism is obscure to us, in some way, the wrap that, when worn by a woman praying or prophesying, honored the head of woman (man) had the exact opposite effect upon the head of man (Jesus), when worn by a man praying or prophesying. Whatever the nature of this ἐξουσία that is based upon creative relationships, it persists, even though man and woman have very different relationships to each other "in the Lord" (1 Corinthians 11:11-12), where both are body parts of Christ (καθὼς καὶ ὁ Χριστὸς τὴν ἐκκλησίαν ὅτι μέλη ἐσμὲν τοῦ σώματος αὐτοῦ Ephesians 5:29-30. Even though man and woman's parallel relationships within Christ give rise to new relationships between them, it seems that the new relationships as being in addition to, rather than instead of, the created ἐξουσία relationship, so that both sets of relationships exist simultaneously. The most straightforward explanation is that, just as the woman rejoins the man's body in marriage, both male and female believers become parts of Christ's body, as if brides in a marriage, so that ἐξουσία exists in two very different, but non-exclusive, relationships at the same time.).
Of course, the fact that the New Testament seems to make better sense with ἐξουσία as ἐκ + οὐσία than as ἐκ + εἰμί carries little weight, apart from suggesting that its authors might have understood and used it as ἐκ + οὐσία.
Ephestion, above, proposed the same etymology as I have, but interpreted it differently. I am not sure that the derivation would make any material difference to the question of whether or not evil rulers have ἐξουσία from God, but it seems clear that there are clear Biblical precepts and precedences supporting disobedience to, and even rebellion against, both religious and civil rulers. However these might be justified, it also seems clear that such cases are to be exceptions to the rule.
The arguments against ἐξουσία as ἔξεστι seem so obvious that it is inconceivable that they have been overlooked. There must be very good reasons for scholars to have reached such counterintuitive conclusions, but they have eluded me. The easiest, but least satisfying, course would be to simply accept the "authority" (ἐξουσία? ἔξεστι?) of noted scholars, without understanding their scholarship on the matter. Perhaps someone can shed more light upon this for me.