The Gemara in Berachos 10a says that (at the advice of his wife Beruriah) Rebbi Meir prayed for neighbors of his to become religious, and the Gemara seems to approve of this. This is also implied by Sotah 14a, where Moshe is said to daven for the wicked to return, as well as in Taanis 23b, where Abba Chilkiyah says that his wife is more righteous because she prays that the wicked should repent. Additionally, R. Yehudah Hachasid (Sefer Chassidim no. 76) and R. Avraham ben Harambam (Hamaspik LeOvdei Hashem, ch. on bitachon) both advice praying for the wicked to repent as proper things to do (though R. Avraham may have a few caveats)

The Maharsha on Brachos asks, 'how is it possible to pray for someone to do teshuvah, isn't that a violation of their free will'? He himself doesn't give an answer, but R. Moshe Feinstein, in a letter to R. Efraim Greenblatt (Igros Moshe O.C. IV 40:13), mentions an idea from Rav Greenblatt that prayer can indeed 'turn' a person, or change his will, so to speak, and since it's coming through prayer of earthly beings that isn't a violation of הכל בידי שמים חוץ מיראת שמים because it isn't 'from Heavan', it's from earth. A similar explanation is given by the Chazon Ish (in a note printed towards the end of Orach Chaim).

R. Moshe Feinstein himself, however, rejects this interpretation, insisting that nothing can override the ability for free will. Therefore, he explains the Gemara that supports praying for others to repent as a prayer for them to either not be tested or for them to hear inspiring mussar, which will likely inspire them to repent. The actual repentance and yiras shamayim, however, is all up to the individual. This online article also quotes the Shu"t Meil Tzedaka, R. Yonah Landsofer, who has a similar view to that of R. Moshe.

The Gemara in Sotah 14a also implies that Mosheh prayed for others to repent, and Sefer Chareidim (Teshuvah Ch. 5) writes that it is a mitzvah to do so. Rav Ovadiah Yosef (Yalkut Yosef O.C. Vol 5 pg. 64) quotes a story regarding the Arizl, who told the Alshich that he should pray for his son, who had apostatized, to return to Judaism. Not only that, but he seems to have advised everyone that it would be appropriate to pray for others to become more religious.