She is able to cry on demand simply by reflecting that summer is going to end. She allows Matsubue to drink her tears, and reflects that she will keep fulfilling his odd request until she figures out his mystery. She wonders why she fell in love with him.
She believes that if she solves his mystery she will be able to solve her own.
Her full name is actually Arika Togawa (though she is not called Arika in the OVA) and she and Takaomi Matsubue are the stars of another manga by Ueshiba Riichi (the author of Mysterious Girlfriend X) called Discommunication.
Discommunication in turn also spawned a spinoff work called Yume Tsukai, which got its own anime series.