Fuel is Lighter's son (tell me you saw a pun there). The only time when he is actually important in the game is during
Chapter 1, when he is stuck at their house alone during a fire. Flint has to travel through the Sunshine Forest and rescue him. Afterwards, he'll travel with Flint for a little bit and heal his HP by cheering him on in the background. Of course, he might do 1 or 2 hits every once in a while, but it really doesn't do much damage. He is good for grinding with Flint if you think you'll have trouble throughout the game. He, like Lucas and Kumatora, has a form for three years after Duster disappears.
Fuel is only in Mother 3 as a minor character. He is a little important, as he helps start the story a bit, but not much.