The English phrase "abomination of desolation" comes from the King James translation of Matthew 24:15. The Greek phrase is τὸ βδέλυγμα τῆς ἐρημώσεως. As Jesus says, it appears in Daniel 9:27 and Daniel 11: 31. It also appears in 1 Maccabees 1:54.
Daniel 9:27 uses the concept after mentioning that it will occur after the Messiah is cut off (yikkaret; Daniel 9:26).
Daniel 11:31 says that it will come after the sanctuary of the temple is polluted and the daily sacrifice is taken away.
The mention in 1 Maccabees 1:54 is perhaps more interesting because the use of the phrase is not a prophecy but rather in a historical text. The situation described is clearly depicted as a fulfillment of the prophecy in Daniel. (Jesus projects it as still in the future.) Prefaced to this use is a description of the actions of Antiochus IV Epiphanes.
Antiochus wanted a unified populous (1 Maccabees 1:41) and wanted his empire to go in a new direction, therefore he commanded "each to abandon his customs" (1 Maccabees 1:42). As a consequence, "many from Israel were happy to serve him and they sacrificed to idols and violated the Sabbath" (1 Maccabees 1:43) and "to follow the customs of the heathen" (1 Maccabees 1:44). They stopped the sacrifices (1 Maccabees 1:45), and "polluted the holy things" (1 Maccabees 1:46). They built altars to idols (1 Maccabees 1:47), "so that they forgot the law and changed all the judgments and whosoever would not do according to the command of the king was put to death" (1 Maccabees 1:49-50). Furthermore, they set up overseers (ἐπισκόπους, "bishops" although "spies" might be a better translation in the context) to insure that everyone followed the new way of thinking. "Everyone who abandoned the law and did evil in the earth" was promoted into positions of responsibility (1 Maccabees 1:52). And then, "on the fifteenth of Chaseleu, in the one hundred and forty-fifth year, an abomination of desolation (βδέλυγμα ἐρημώσεως) was established at the altar" (1 Maccabees 1:54).
This passage shows the processes that led the authors of 1 Maccabees to decide that the prophecy of Daniel had been fulfilled. Polluting the sanctuary and cutting off the daily sacrifices are both mentioned to draw the reader's attention that the conditions had been met. But following the Gentile customs and doing things their way. Covenants and the law were both abandoned. Spies were put in place to report and insure that everyone behaved the way that the new regime wanted them to. And "they burned the books of the law which they found" (1 Maccabees 1:56). It was, after all, important to rewrite the history and make the previous orthodoxy disappear.
This description from Maccabees supplies an understanding of what Jesus was predicting for the future of Christianity before his death.