In a small bowl, cream butter and sugars until light and fluffy. Beat in egg and vanilla. Combine the flour, oats, baking soda and salt; gradually add to creamed mixture and mix well. If too wet, add more oatmeal as needed. Stir in cherries.
Drop by healthy Tablespoon or rolled into balls 3 in. apart onto ungreased baking sheets. Bake at 350° for 14-16 minutes or until golden brown. Cool for 1 minute before removing from baking sheets to wire racks. Store in an airtight container.