According to the search logs, it's season for both. Here's a bunch of recipes that use both. The common third ingredient is citrus, either lime or lemon, either in the recipe or served as the with.
Lemon cupcakes with blueberry basil filling. from the blog A Good Appetite:
Blueberry Basil Sauce 1/3 cup frozen or fresh blueberries 1 T water 1/2 t lemon juice 1 1/2 T granulated sugar 6 large basil leaves, chopped
Blueberry, basil and rum cocktail, Art of Drink:
Muddle basil, mint, sugar and lime juice in base of a shaker glass. Add blueberries and continue to muddle. Add rum and ice and shake. Fill collins glass with ice and strain muddled rum mixture into the glass and top with club soda. Garnish with a lime.
Blueberry basil ice cream, Washington Post:
Combine the blueberries, basil leaves and 1/4 cup of the sugar in a small saucepan over medium-low heat. Cook for about 10 minutes, stirring frequently, until the berries collapse and yield their juices. Transfer to a blender and let it cool there for 30 minutes.
Grilled chicken with blueberry basil salsa, Martha Stewart:
Place 2 cups blueberries in a food processor; pulse until coarsely chopped, about five times. Transfer to a medium bowl, and add onion, jalapenos, lime juice, and 1/4 teaspoon salt; stir to combine. Add basil, cilantro, remaining cup blueberries, and oil; stir to combine.
Given this collection, you should be able to create many more variations.