Put the onion chunks in the food processor; pulse to chop finely. Scoop into a sieve, rinse under cold water, and set aside to drain. Put the cucumber chunks in the food processor; pulse to chop finely. Add the drained onion and pulse to combine. Set aside 3/4 cup of the cucumber-onion mix to stir into the soup at the end. Cut the avocado in half and remove the pit. Scoop out the pulp and drop it into the food processor with the remaining cucumber-onion mixture. Add the yogurt, olive oil, garlic, cumin, and salt. Puree until fairly smooth. Using a sharp knife, slice the basil and mint leaves into shreds and then stir them into the soup along with the lemon juice, 1 cup cold water, and the reserved cucumber-onion mixture. Season to taste with salt and pepper. Cover and chill for several hours. Before serving, check for consistency, gradually adding another 1/2 to 1 cup cold water to thin the soup if necessary. Add more lemon juice and salt if needed and serve.