She wants people to remember her birthday so she reminds them by "dropping hints" - that is unsubtle reminders to her friends which don't necessarily refer directly to her birthday but are clearly such a reference. For example, she could say to her friend, " I really liked your birthday party" as a hint to her friend that her birthday was soon.

Drop a hint means to suggest something to someone. She keeps suggesting things that she wants to happen for her birthday. or gifts she wants? i don't know. I'd need more context.