There is a custom that originates from the Arizal (Rabbi Yishak Luria of Safed, 1534-1572) to pray during the month of Elul for peers or family members who have, Heaven forbid, strayed from Torah observance. As Elul is a time of divine compassion, when the Almighty is particularly receptive to our prayers, especially with regard to Teshuba (repentance), it is proper to pray during this month that God should bring those who have strayed back to the path of religious observance. The Arizal taught that one should insert this prayer three times a day during Elul – in Shaharit, Minha and Arbit – in the Beracha in the Amida that deals with repentance ("Hashibenu"). Just prior to the concluding blessing of that Beracha, meaning, before one recites, "Baruch Ata Hashem Ha’rose Bi’tshuba," one should recite the following text:
"Yehi Rason Milefanecha Hashem Elokenu V’Elokeh Abotenu She’tahtor Hatira Mi’tahat Kiseh Kebodecha Le’hahazir Bi’tshuba Shelemah Le’khol Posh’eh Yisrael U’bi’chlalam Tahazireni Li [the person’s name] Ben [the person’s father’s name] Ki Yemincha Peshuta Le’kabel Shabim." (Translation: "May it be the will before You, Hashem our God and God of our forefathers, that You shall dig an opening underneath Your Throne of Glory to bring all sinners of Israel back in full repentance, and among them bring back for me so-and-so son of so-and-so, for Your right [hand] is outstretched to receive those who return.")
There is also an admirable custom to spend additional time each day during the month of Elul studying works of Musar (religious exhortation). Recommended works to learn include Hobot Halebabot by Rabbenu Bahya Ibn Pakuda (Spain, 1050-1120), Reshit Hochma, and the third section of Shaareh Teshuba by Rabbenu Yona Gerondi (Spain, 1180-1263). This section offers important religious guidelines and deals with the various Misvot and prohibitions regarding which one must exercise particular care, and it is thus especially relevant to the period of Elul. Some also advise studying the work Orhot Hayim, which is attributed to the Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327).
Summary: It is proper during the month of Elul to pray in the Amida on behalf of those who have strayed from Torah observance. This prayer should be added in the "Hashibenu" section of the Amida. It is also proper to spend extra time each day during Elul studying the classic works of Musar.