According to Mori V’Rabi, Harav Hagaon R.Shlomo Miller Shlita, naming a child after a non-Frum family member is acceptable & at times even recommended.
The main reason we name after an individual who passed away is to honor him or his living relative. It is usually done as הכרת הטוב. (Gratitude)
הכרת הטוב overrules all the reasons for not naming after a non-Frum person.
We find בני ישראל as a הכרת הטוב named their children after Alexander the Goy
This part is from ספר יוסיפון. It is a continuation of the story in the (גמרא (יומא ס"ט ע"א.
took Alexander on a tour of the .בית המקדש Alexander, wished
to donate gold to have an image of himself placed in the בית
המקדש. שמעון הצדיק told him that it was forbidden for
the Yidden to have images, and certainly not in the בית
המקדש. He suggested that he instead give the gold to the poor. And as
for memorializing the occasion,שמעון הצדיק suggested
an even better way: all-male כהנים born
that year would be named “Alexander.”