Goomnuts are acorn-like nuts in Paper Mario that Mario gets from hitting the tree in Goomba Village with a Hammer. They heal 3 Heart Point alone. They can be cooked into a Nutty Cake.
One of these trees is also in the background of the fight against the Goomba King. If you hit the tree a huge goomnut will hit the King, Red Goomba, and Blue Goomba. Although if the king-hits his side 3 goomnuts hit you.
In Luigi's Diary, he put that Goombario's family came to visit. He mentioned that they gave him a Goomnut and his Flower Point increased by 3. About Mario, he says, "Do you think he will notice?"