Preheat oven to 350 degrees. Butter an 8-inch square baking pan. Line with parchment paper, leaving a 2-inch overhang on two sides. Butter and flour parchment and pan, tapping out excess flour.
Make streusel: Whisk together butter, brown sugar, and salt. Add flour and mix with a fork until large crumbs form. Refrigerate until ready to use.
Make cake: In a medium bowl, combine rhubarb, brown sugar, and 1/4 cup flour. In another medium bowl, whisk 3/4 cup flour, baking powder, and salt. In a large bowl, using an electric mixer, beat butter and confectioners' sugar until light and fluffy; beat in eggs, one at a time. With mixer on low, beat in vanilla, then flour mixture. Spread batter in prepared pan. Sprinkle with rhubarb and top with streusel.
Bake cake until golden and a toothpick inserted in center comes out with moist crumbs attached, 45 to 50 minutes. Let cool completely in pan. Using paper overhang, lift cake from pan. Cut into 16 bars.