Place the vegetable oil into a large pot over medium heat. Stir in the sliced onions, and cook until transparent, about 5 minutes. Add the tomatoes and tomato sauce, using a spoon to slightly chop the whole tomatoes. Season with 1 1/2 teaspoons salt and 1/4 teaspoon pepper. Cover, and simmer for 20 minutes, stirring occasionally. Stir the cabbage into the tomatoes. Cover, and simmer 1 hour until the cabbage is tender and mixes easily with the tomatoes.
Place the ground beef, eggs, finely chopped onion, rice, water, additional 1 1/2 teaspoons salt, and 1/4 teaspoon pepper in a mixing bowl. Mix together with hands until thoroughly blended. Add additional rice if mixture is too moist. Form into golf ball-sized meatballs. Place meatballs on a plate, and refrigerate 20 minutes to firm.
Place meatballs on the surface of the tomato mixture. Cover and simmer over medium heat until the meatballs are firm and cooked through, about 15 minutes. Gently stir the meatballs into the tomato mixture. Cover the pot and continue to simmer 45 minutes more. Stir in the lemon juice, and taste for seasoning, adding additional lemon juice if required. Stir in the raisins, and honey. Add the brown sugar, a little at a time, tasting after each addition. Cook mixture over medium heat 15 minutes more.