Combine fruit, water, sugar, and ginger in a saucepan. Bring to a boil, then reduce to a simmer for about 15 minutes. Once the fruit is soft, use a potato masher to crush the fruit. Add the lemon and allow it to simmer for 5 more minutes, then turn off the heat.
Once it's room temperature, pour the juice through a fine-mesh strainer and refrigerate until ready to use.