Giving a birthday gift to a girlfriend, like giving a gift to a girlfriend in other days, is to express the sentiment of "I love you". Of course, it is better to give a gift to a girlfriend that is suitable for her. Since your girlfriend is frugal, you can give her some things that she usually lacks, whether it is food, clothing or daily necessities. In terms of food, of course, you must have a birthday cake, and then take her to a candlelight dinner. You don't have to go to a too expensive restaurant. As long as you celebrate your birthday together, I believe she will be very happy.
In terms of clothing, according to the season, see what clothes your girlfriend lacks, or what clothes she likes but is not willing to buy. If you buy them for her on her birthday, I believe she will be very touched.
There are also things used in daily life, such as jewelry, cosmetics, cooking utensils, etc., as long as they are what she needs, giving such gifts is practical and will not make her think you spend too much.