This appears to be a British expression. The word button, in this expression, may mean the bud of a flowering plant before it blooms (before the flower bud opens. Buttons (buds) are usually small, so the expression means 'something or someone that is cute and small'. This expression is often used to describe a beautiful, young baby, e.g., "Oh, your baby is as cute as a button".

It's just an expression. It just means really cute. Has nothing to do with buttons.