This boy's name is of Old English origin, pronounced RAM-zee, has a few possible meanings. One is the seemingly obvious "ram island," while another is "raven island." In Old English, hræm was raven, and ramm was ram. In Old Norse, rammr was strong, giving this a third meaning of "strong island."
The name comes from the Clan Ramsay, dating back to the 11th century, beginning with Symon de Ramesie. As a surname there are dozens of namesakes, but the spelling Ramsey is more popular for namesakes as a first name. The Baron de Ramsey is also a title in the Peerage of the UK. It seems that the spelling Ramsey comes as a toponymic surname.
The spelling Ramsay was given to only 7 boys in 2016, hinting that it is still primarily a surname, while Ramsey was chosen for 150 boys, definitely falling in the category of place-name turned baby name.
Perhaps the only well-known literary & TV character is Ramsay Snow (aka Ramsay Bolton) from the Game of Thrones series. This character, however, is not one you would want to name a child after. I would suggest hunting down other, positive, references. One is jazz musician Ramsey Lewis, another is political activist and 66th Attorney General Ramsey Clark.