Betta's actually need a solid cover..so that the temperature of the air above the water doesn't get colder then the water itself. They come up to breathe air quite often..breathing cold air (like air conditioned air) can make them quite sick. Having a solid cover will also keep the water temp from fluctuating quite so much.
Using cardboard will only work for so long..as the humidty that builds up inside the bowl will make the cardboard mildew, which can lead to bacterial infections, or worse.
You're best bet it to use something made of solid plastic (like either plexiglass or a bowl lid..some poepel have even used cd's for their bowls) and then cut a small hole in it. You could silicone some of the plastic mesh on the bottom side, if you liked, to ensure that your betta does not jump out of the hole, as they are famous for being able to sqeeze through small gaps.