I found different years cited as Buddha's birth year. They were 560, 563, 564, 623 and 624 BCE. Almost all sources cited the full moon of May as the birth date, although 2 sources cited April 8.
The May full moons for these years are: May 7, May 11, May 21, May 14 and May 25. April 8 was a full moon in 560 BCE.
The Cambridge and Oxford histories of India accept 483 B.C as the date of Buddha's nirvana. He was 80 years old when he died, so this puts his birth year at 563 BCE.
If we accept 563 BCE as Buddha's birth year, his birth date would be May 11.
However, in January 2006 an Indian historian challenged the historical assertion that Buddha was born between 560-624 BCE. He cites evidence that Buddha was born in 1887 BCE. Assuming he was born on the May full moon, his date of birth would be May 8, 1887 BCE.