I'm a woman who wakes up in the morning to pray before Fajr but during my period (which stays around 9 days) I don't do any prayers. What can I do during those days? Can I study Qur'an by heart, by reading Qur'an from the computer? Can I wake up before Fajr and still do something to please Allah like saying (Istighfar)?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Some Muslim scholars are of the opinion that reciting the Qur'an is permissible for a menstruating woman. This is the opinion of al-Malikiyah and one of the opinions of Imam al-Shafi'e . It is also reported from Imam Ahmad . However, the majority of Muslim scholars are of the opinion that reciting the Qur'an during the menstruating period is not permitted. The preponderant opinion - Allah knows best - is that reciting the Qur'an by heart, or by computer, or the Qur'an without touching it directly is permissible.
Moreover, it is also permissible for such a woman to remember Allah by asking His forgiveness, invoking Takbeer, saying La Ilah Illa Allah, making Tahmeed, supplications of the morning, evening, sleeping, upon entering house or going out, etc. Likewise, it is also permissible for her to awake before Fajr and implore Allah's forgiveness.
Allah knows best.