I guess that you must not get enough and good quality sleep during period that cause puffy eyes. Or drinking too much water at night can also be the factors of puffy eyes. If so, don't worry, the puffy eyes will go as long as you get enough sleep and good life style. Right now, you shall try to reduce the puffy eyes if you want to get a good look. Commonly, Cold compress are recommended to people to reduce puffy eyes. But since you have your period, maybe you can try warm compress which can also used to reduce puffiness.
Well, I guess it may be caused by your eating too much salty food during your period. At this time, there is too much water stored inside your body, so naturally, your puffy eyes just come into being. But I have some home remedies for you to remove your puffy eyes. Apply a cool compress to your eyes and your situation will get much better, trust me. Or you can also place an ice cold spoon on your eyelids, which is said to be very effective. You can try this.