What are common explanations for the angel being delayed for 21 days in Daniel 10:13?
This is not a common explanation, but I believe it is the correct explanation:
When you read the chapter and read in verse 1 that Daniel "understood the thing, and had understanding of the vision" (it is repeated for emphasis) you must wonder, like I have done many times in the past, "What on earth is there to understand? There seems nothing much here!" Then a few days ago I thought What if the 21 days delay is part of the message?
Just as Jeremiah was told how long the Babylonian Empire would last, 70 years, so Daniel was told how long the Persian Empire would last. The difference is that the Holy Spirit chose to keep the duration of the Persian Empire hidden in a vision: this vision is recounted in Daniel chapter 10. Daniel himself understood the vision (verse 1), but the Holy Spirit did not want him to explain it in the word of God.
The 21 days represent the 210 years starting with the Fall of Babylon 539 bc and terminating in the death of the last Persian Emperor Artaxerxes V, i.e. Bessus, at the hands of Alexander the Great in the summer of 329 BC.
There may be another explanation instead of or in addition to the above.
Daniel had this vision "in the third year of Cyrus" (Dan 10:1). In his first year he issued a decree that the Temple in Jerusalem be rebuilt (Ezra 1:1-2).
[The "Cyrus Cylinder", a copy of which is in the entrance vestibule of the United Nations HQ, was issued for a different temple and is strongly suggestive he issued the same decree for all temples throughout his empire.]
The date the third year of Cyrus is dated from his conquest of Babylon which fell on the night of 12th Oct 539 BC. According to Babyonian reckoning his Accession Year was from Oct 539 to Nisan 538. His first year started Nisan 538, and his third year began Nisan 536. It was in this year that Daniel had the vision.
Why was Daniel "mourning three full weeks"? Maybe because he had heard that the temple had begun to be rebuilt, but had been brought to a halt by the enemies of the Jews. The delay of 21 days may be a declaration by God that the temple would not be finished for another 21 years, a year per day. The temple was finished on 3rd Adar in the 6th year of Darius I (Ezra 6:15), which according to Parker & Dubberstein's "Babylonian Chronology - 626 BC to AD 75" was 12th March 515 BC (Julian).
The Jews used inclusive counting. In this case it means that the remainder of Cyrus's third year after the time of the vision is counted as a whole year.
So the babylonian year containing 12th March 515 BC (when temple was finally completed) began in Nisan 516 BC which using Jewish inclusive counting was 21 years after Daniel's vision in the year starting Nisan 536 BC.
The delay of 21 days is God's way of telling Daniel the temple would not be completed for another 21 years, which Daniel understood, but the Holy Spirit did not direct Daniel to reveal it to the reader explicitly.