お世話になる is a set phrase said to someone you expect to be in contact with from now onwards, and it means kind of "let's look after each other" (see the basic meaning of お世話する). For example, it is typically said to a neighbour when you move in to a new apartment.
Conversely, when you move out and you don't expect to be in touch frequently with this person, you can say お世話になりました (note the past tense), to appreciate that this person has been there for you until this moment.
Therefore, taking a look at your context, I guess it just means that Daiki is considering to stay a litte bit longer:
Daiki: てっちゃんには言ったじゃん、俺出るって。I've told Techhan [you], that I'm leaving.
Tecchan: うん、きいてたね 俺はね。Yes, I've heard it.
Daiki :でもちょっとまだお世話になろうかな。But, I'm thinking of taking care of each other [staying] a little bit more.