Q. Sometimes the minhag is to name someone using their mothers name, and sometimes using the fathers name. Why is this? what is the rule when? Mothers name only when davening for Refuah?

what about dovening for parnassa? Halzlocha in learning?

   What happens when the parents divorce. does the father daven for his child using his ex wife’s name, or does he change to using the fathers name?
   A. There are a number of sources for using an individual’s mother’s name when praying for one who is ill. Piskei Teshuvos (116: 2) quotes Ponim Yofos that prayers for healing are said using the mothers name, while the ones for remembering the soul of the departed are said mentioning the fathers name. This is based on the Talmud (Shabbos 66b and Brochos 55b) as well as the Zohar (Lech Lecha) and the verse on Tehilim (86: 16), since there is more compassion and spiritual mercy given to the mother. On other matters such as Torah, spiritual or monetary needs etc. the name of the father is used
Horav Shlomo Miller’ Shlit’a opinion is that when the parents divorce, the father daven for his child using his ex wife’s name.
Rabbi A. Bartfeld as advised by Horav Shlomo Miller and Horav Aharon Miller Shlit’a