As your policy will be different when you add a named driver, it will certainly have an effect on your premium. But depending on who the additional driver is, the cost could go up or down.
If, for example, you add someone who is considered a higher risk, for an inexperienced, younger driver such as your teenager who has just passed their test, then it is likely to cost you more.
On the other hand, if you are a young driver and add someone with twenty years of no claims, it’s likely to improve your rate. This is because the overall risk - in an insurer's eyes - has gone down, so they will be happy to lower your costs.