It could be that an ancient artifact in the Naruto world has a connection to the Super Saiyan bloodline. This artifact was once used by a powerful ninja who had some sort of encounter with a Saiyan in the past. When Naruto or another ninja discovers this artifact, it unlocks the latent Super Saiyan bloodline within them. So, they gradually start to show Saiyan traits like the golden hair and the intense power - up when angry.
One way could be through some sort of cross - dimensional rift. Maybe a wizard or a powerful jutsu accidentally opens a portal to the Dragon Ball universe where the Super Saiyan bloodline exists. Then, a character in Naruto, perhaps Naruto himself, gets exposed to this bloodline energy and starts to develop Saiyan - like abilities, such as increased strength, speed, and the ability to transform.
There could be a race in the Naruto world that has a dormant Super Saiyan - like bloodline. This race was long forgotten or hidden. Through some plot - driven events, like a search for a powerful power source to defeat a new and dangerous enemy, Naruto and his friends discover this race. Interaction with them causes the activation of the Super Saiyan - like bloodline in Naruto, which then changes the power dynamics in the Naruto universe.