The Collaborative International Dictionary of English v.0.48:

Dilemma \Di*lem"ma\, n. [L. dilemma, Gr. ?; di- = di`s- twice +
   ? to take. See Lemma.]
   1. (Logic) An argument which presents an antagonist with two
      or more alternatives, but is equally conclusive against
      him, whichever alternative he chooses.
      [1913 Webster]

   Note: The following are instances of the dilemma. A young
         rhetorician applied to an old sophist to be taught the
         art of pleading, and bargained for a certain reward to
         be paid when he should gain a cause. The master sued
         for his reward, and the scholar endeavored to elude his
         claim by a dilemma. "If I gain my cause, I shall
         withhold your pay, because the judge's award will be
         against you; if I lose it, I may withhold it, because I
         shall not yet have gained a cause." "On the contrary,"
         says the master, "if you gain your cause, you must pay
         me, because you are to pay me when you gain a cause; if
         you lose it, you must pay me, because the judge will
         award it." --Johnson.
         [1913 Webster]

   2. A state of things in which evils or obstacles present
      themselves on every side, and it is difficult to determine
      what course to pursue; a vexatious alternative or
      predicament; a difficult choice or position.
      [1913 Webster]

            A strong dilemma in a desperate case!
            To act with infamy, or quit the place. --Swift.
      [1913 Webster]

   Horns of a dilemma, alternatives, each of which is equally
      difficult of encountering.
      [1913 Webster]

WordNet (r) 3.0 (2006):

dilemma
    n 1: state of uncertainty or perplexity especially as requiring
         a choice between equally unfavorable options [syn:
         dilemma, quandary]

Moby Thesaurus II by Grady Ward, 1.0:

79 Moby Thesaurus words for "dilemma":
   Aristotelian sorites, Goclenian sorites, Gordian knot,
   Scylla and Charybdis, baffle, bafflement, bewilderment, bind,
   bother, box, catch-22, categorical syllogism, choice of Hercules,
   confoundment, confusion, corner, crux, deadlock, difficulty,
   discomposure, disconcert, disconcertedness, disconcertion,
   disconcertment, disturbance, embarrassment, enigma, enthymeme,
   figure, fix, hole, impasse, jam, knot, knotty point, mode,
   modus tollens, mood, mystery, mystification, no choice, node,
   nodus, nonplus, only choice, oxymoron, paradox, paralogism,
   perplexity, perturbation, pickle, plight, pons asinorum, poser,
   pother, predicament, problem, prosyllogism, pseudosyllogism,
   puzzle, puzzlement, quandary, riddle, rule, rule of deduction,
   scrape, snooker, sorites, spot, squeeze, stalemate, stew, stymie,
   syllogism, teaser, trouble, unassuredness, upset, vexed question