Melt butter in an 8-qt. saucepan over medium-high.
Cook onion until soft, 5–7 minutes.
Add blueberries, jam, vinegar, cinnamon, ginger, cabbage, apple, salt, pepper, and ¼ cup water; bring to a boil.
Reduce heat to medium; cook, covered, until cabbage is tender, about 1 hour.
