The angel Moroni appeared to Joseph Smith on the night of September 21, 1823 (this happened to be a Sunday). This happened to be the third day of the Jewish holiday Sukkot, which is a week long holiday to commemorate the 40 years that the children of Israel spent in the wilderness.
After showing the gold plates to Joseph Smith, the angel Moroni instructed him to return to where the gold plates were buried in exactly one year's time. Joseph returned to the place on September 22 for the next four years, until Moroni finally allowed Joseph to get the plates, in 1827.
In 1827 the Jewish holiday of Rosh Hashanah happened to fall on September 22 (or from sundown on the 21st until sundown on the 22nd). The holiday of Rosh Hashanah is the Jewish new year but it is not called Rosh Hashanah in the Bible, instead it is referred to as "a memorial of blowing of trumpets". By tradition it is a day of judgement when the books of life are presented before the Lord and everyone is judged "by the books" so to say. The righteous are sealed up to eternal life, the wicked are sealed up to destruction and everyone else is given 10 days to repent when final judgement is pronounced on Yom Kippur, which is the day of atonement.
So we have a holiday that is celebrated by blowing a horn, the books of life are presented out of which men are judged, but a space is given for men to repent before final judgement on the day of atonement. And it was on that day that the angel Moroni chose to give the Book of Mormon to Joseph Smith. We have picked up some of this symbolism with how we portray Moroni on our temples, but perhaps he should have a rams horn and not a straight trumpet.
We associate Moroni with the angel spoken of in the Book of Revelation:
6 And I saw another angel fly in the midst of heaven, having the everlasting gospel to preach unto them that dwell on the earth, and to every nation, and kindred, and tongue, and people,In the revelation the next angel to follow proclaimed the downfall of Babylon. But between those two angels the gospel must be preached and carried to all nations. Only then will Babylon be overthrown.
7 Saying with a loud voice, Fear God, and give glory to him; for the hour of his judgment is come: and worship him that made heaven, and earth, and the sea, and the fountains of waters.
I like to think that Moroni chose that date to fulfill the Law and for the significance that it had for him even if Joseph Smith didn't know the significance of the date.