Reduce the heat to low, remove the lid and use a potato masher or the back of a fork to mash up about at least half of the beans. Continue to cook the beans, uncovered, stirring often, for 3 more minutes. Remove the saucepan from the heat and stir in the salt, pepper and lime juice. Taste and add more salt, pepper, lime juice if necessary. If the beans seem dry, add a very small splash of water and stir to combine. Cover until you're ready to serve.
To cook the black beans: In a medium saucepan over medium heat, warm the olive oil until shimmering. Add the chopped onion and a sprinkle of salt. Cook, stirring occasionally, until the onion has softened and is turning translucent, about 5 to 8 minutes. Add the cumin and garlic. Cook, stirring constantly, until fragrant, about 30 seconds. Pour in the drained beans and water. Stir, cover and cook for 5 minutes.
To make the pico de gallo: In a small mixing bowl, combine all of the ingredients. Stir to combine, and set the pico de gallo aside to marinate while you make the remaining components.