As a serious Harry Potter fan who has read this book many times, I can tell you that there is no great significance to the number three found in the first book of the series. Once Harry has met his friends Ron and Hermione, they are a group of three who experience all the adventures of their first year at school together. Draco and his closest friends also move in a group of three.

Later on in the series, when we are introduced to the unforgivable curses, we find that there are three of those as well. These are not introduced in the first book. These curses include avada kedavra, which is the killing curse, crucio, which allows the person casting the spell to torture and cause unspeakable pain to the victim, and imperio, which forces the person being cursed to do whatever their attacker wishes them to do.

Much later in the series, when Harry discovers the legend of the three Peverell brothers, he also learns that there were three deathly hallows: the elder wand, the cloak of invisibility and the resurrection stone.

If I were to pick a number that has significance throughout the Harry Potter series, I'd pick seven. There are seven books in the series, seven siblings in the Weasley family, and seven horcruxes.