Preheat oven to 350 degrees. In a food processor, process graham crackers, 2 tablespoons sugar, and 1/4 teaspoon salt until fine crumbs form. Transfer to a medium bowl and stir in melted butter until crumbs are moistened. Wipe processor bowl clean. With a flat-bottomed 1-cup measure or glass, press crumbs evenly into a 9-by-13-inch baking pan. Bake until crust is firm, 15 minutes. Let cool on a wire rack. Reduce heat to 325 degrees.
In a small saucepan, bring apricots, 1/4 cup sugar, and pinch of salt to a boil over medium-high, stirring, until sugar dissolves. Reduce heat and simmer, stirring frequently, until mixture is shiny, 10 minutes. In food processor, puree apricot mixture, lemon juice, and 1 tablespoon water.
In a large bowl, using an electric mixer, beat cream cheese and sour cream on medium until smooth. Add 3/4 cup sugar and beat until smooth. Add vanilla and pinch of salt and beat to combine. Add eggs and beat until smooth, scraping down bowl as needed. Pour cream-cheese mixture into cooled crust and smooth top.
Randomly drop small spoonfuls apricot puree on cream-cheese mixture. With a skewer or thin-bladed knife, gently swirl puree. Bake just until set, about 25 minutes. Let cool slightly in pan on rack, then refrigerate until chilled and firm before cutting into squares, 2 hours.
Cheesecake can be refrigerated, covered with plastic wrap or foil, up to 2 days.
Change it up: Substitute 3/4 cup strained jam for the apricot mixture -- we like blueberry.