Put onion in a small bowl; add water to cover. Let soak for 10 minutes. Drain well; pat dry with a paper towel.
Prepare tofu: Open tofu and invert onto a plate. Use paper towel to blot away excess liquid. If using medium tofu, drain on lint-free towel or 2 stacked sheets of paper towel on a plate.
Cut medium or block tofu into 1-inch cubes or 1-inch-thick dominoes; arrange in a shallow bowl or individual dishes. (If using tofu pudding, scoop up large shards with a metal spoon, putting into shallow bowl or individual serving dishes.) Before serving, pour off any whey.
Drizzle with soy sauce. Top with green onion, ginger, and optional garnishes (or leave tofu naked and let diners choose their own toppings).