Put beet in a small saucepan with water to cover by 1 in. Bring to a boil, then simmer until tender when pierced with the tip of a small knife, about 25 minutes. Drain. When cool enough to handle, peel and cut into quarters.
Meanwhile, put ginger in a medium saucepan with water to cover by 2 in. Bring to boil. Drain and repeat process once or twice more, depending on the spiciness of the ginger. It should have a strong taste but not be too spicy.
Combine rice vinegar and salt with 1 cup water in a small bowl, stirring to dissolve salt. Add ginger and beet, cover, and chill overnight.
Make ahead: Up to 2 weeks, chilled.
Note: Nutritional analysis is per tbsp.