The OC could be introduced as a new student arriving at Hogwarts just before the start of the school year. They could make a grand entrance on the Hogwarts Express, catching Harry's attention when they perform a small but impressive bit of magic.
The OC might be introduced during a Quidditch match. They could be a new player on one of the opposing teams. After the match, Harry and the OC have a conversation where they realize they have some common interests, like a love for a particular type of magical creature, and this starts their relationship within the fanfiction.
Another way is to have the OC be a relative of one of the existing characters. For instance, they could be a distant cousin of Hermione who comes to visit and ends up getting involved in the events of the Goblet of Fire. They gradually form a friendship with Harry as they are both interested in solving the mysteries surrounding the tournament.