My question is about Du'a. I wonder how to make Du'a for a special person. Can I do it during Salat and how should I do it? Can I say his name or how?

All perfect praise be to Allaah, The Lord of the Worlds.  I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

Yes, it is permissible to supplicate during the prayer for a specific person mentioning his name. Imaam Al-Bukhari reported that Abu Hurayrah narrated that the Prophet, sallallaahu ‘alayhi wa sallam , once supplicated in the last Rak’ah (unit) of the 'Ishaa' prayer, saying: “O Allaah, rescue 'Ayyaash Ibn Abu Rabee'ah; O Allaah, rescue Al-Waleed Ibn Al-Waleed, O Allaah, rescue Salamah Ibn Hishaam.”

Therefore, this is evidence for the permissibility of supplicating for someone by name.

For instance, one can make the supplication when prostrating, saying, "O Allaah, forgive so and so," or after Tashahhud (the prayer said when seated in prayer) and before concluding the prayer.

Allaah Knows best.