Etymologically it means the time just before the dawn, which is why it can't mean "tomorrow" in a general sense. Morgen as a noun means morning, but as an adverb means tomorrow, which is similar to many other languages' use of the word for morning. (FYI, the etymology of tomorrow is from Old English "tomorgen" which obviously means on/at/to the morning.)
Am I correct in assuming that in Dutch 'de nacht' does not have quite the same semantic sense as in English? As when an English speaker might say 'I went to a party last night (meaning evening, not meaning all through the night into the wee hours). In that case would one most likely use 'de avond' in Dutch?