Writing Names for Elevation of the Soul and Blessing

Hello, Rabbi! We are engaged in publishing books, and we have questions regarding dedications (for the elevation of the soul) as well as thanks to living people. The question concerns the names that should appear in dedications: should it be the father's name or the mother's name? This confuses us a bit because in the 'Mi Sheberach' prayer when called to the Torah, we mention the mother's name. Also, when we mention the deceased, it seems to me that we always mention the mother's name and not the father's. Perhaps there is a different custom among Ashkenazim, which is why they gave us dedications with the father's names and not the mother's, and this confused us a bit. I would greatly appreciate a full answer to this question. How should the name of the deceased be correctly indicated (name and mother's name or name and father's name)? And how should the name of a living person be correctly indicated? Also, I am interested to hear what Rabbi Ovadia Yosef said on this matter.

Shalom Rav 

Prayer for living people [for blessing, healing, success, marriage, etc.] - it is appropriate to pray using the mother's name, but it is not mandatory. 

Reciting 'Mi Sheberach' after being called to the Torah - the Ashkenazi custom is to mention the father's name. Some Sephardim mention the mother's name, as in any prayer for success, as mentioned above. 

In prayer for the deceased, as well as dedications for the elevation of the soul, etc. - the Ashkenazi custom is to mention only the father's name. Among Sephardim, there are different customs. Each custom has its sources. 

Similar things were written by the Gaon, Rabbi Ovadia Yosef [Shut Yabia Omer, part 2 - Orach Chaim, siman 11].

Gemara Shabbat 66: 'in the name of the mother'. 

Chochmat Shlomo there, that the reason is that lineage through the mother is more evident. 

Zohar Chadash Lech-Lecha 84.