I found here some suppositions:

"The child's name was unusual and unexpected. "Archie" was variously described as a tribute to Meghan's dead cat, a tribute to a military commander Harry served with in Afghanistan, and an anagram of Meghan or Charles' names. It was also the name of a character within the sad life of the first Duke of Sussex, a redhead who married against his parents' wishes. Archibald was the name of his wife's lover, the man who was probably the real father of the Duke's children."

[link to nuttyflavor88.blogspot.com]