Purée the fruit in a food processor until it’s completely smooth, 2 to 3 minutes, scraping down the sides occasionally. If using strawberries, strain the purée through a fine sieve to remove the seeds (press hard with a rubber spatula to extract as much liquid and pulp as possible).
Pour the fruit purée into a medium saucepan and bring to a boil over medium heat, stirring occasionally. Reduce the heat to a simmer and cook, stirring often and scraping the sides and bottom of the saucepan with a heatproof spatula, until any foam subsides and the purée turns glossy, 8 to 12 minutes for blueberry, 20 to 25 minutes for apricot, 35 to 40 minutes for cherry, or 35 to 55 minutes for strawberry. You should have about 1-3/4 cups.
Remove the saucepan from the heat and stir in the honey to taste. (The purée can be refrigerated for up to 3 days at this point. Cover tightly, placing plastic wrap directly on the surface of the purée. Bring to room temperature before proceeding.)
Position a rack in the center of the oven and heat the oven to 170°F, using a convection mode, if available. Line a large rimmed baking sheet (18x13 inches) with a nonstick baking iner, such as a Silpat (16-1/2 x 11-5/8 inches). Pour the purée down the center of the baking liner and spread it evenly with a large offset spatula to a 1/8-inch thickness, leaving a 1/2-inch border all around. The edges should be slightly thicker than the middle, as they dry out faster.
Dry the purée in the oven, rotating the pan halfway through, until it looks leathery and feels somewhat firm to the touch but is still tacky and not completely dried, 6 to 7 hours (or 3-1/2 to 4-1/2 hours on a convection setting). Transfer the baking liner to a cooling rack and let the leather sit at room temperature until it separates easily from the baking liner and is mostly dry on top, 24 to 72 hours. The strawberry leather may still be slightly tacky,and the blueberry leather may have a bit of moisture on top—blot it with a paper towel before wrapping.
Place a piece of parchment over the leather. Flip the leather parchment side down and peel off the baking liner. Starting with a long side, roll the leather and parchment into a cylinder. Using kitchen shears or a sharp knife, cut the leather into 1-inch segments. Wrap each segment in plastic wrap and store them in an airtight zip-top plastic bag at room temperature.
Fruit leather can be stored in an airtight plastic bag at room temperature for 3 to 4 weeks or in the freezer for 2 to 3 months.