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Answer: We have not found a halachic discussion of this interesting point. We will start by understanding the practice of using the mother’s name for tefillot. There are possible allusions to this in Chazal. In Shabbat (66b), Abaye quotes his adoptive mother as saying, according to Rashi’s explanation, that incantations should use the person’s mother’s name.
The gemara in Berachot (55b), describing steps to take when one is in a certain precarious situation, cites a declaration, including "I, ploni son of plonit (according to some texts of the gemara)." Some explain (see opinions in Yabia Omer II, Orach Chayim 11) that we are more likely to know for sure who one’s mother is than who his father is (apparently, we do not want to take chances). The Sifra (Emor 1:5) uses this distinction to explain why the Torah mentions both parents when allowing a kohen to take part in their burial. The Ben Yehoyada (Berachot 55b) considers that "concern" a disgrace to one’s father and gives several areas, spiritual and physical, in which a mother’s impact on her child is greater than a father’s, as well as the contention that a mother is likely to have fewer spiritual liabilities. The Panim Yafot (Bamidbar 12) sees Moshe’s mention of a baby coming out of his mother’s womb in his prayer for Miriam as inspiration for using a mother’s name in prayers.
Yabia Omer (ibid.) posits that all of the above can only create a preference for our formula, but that it does not make a true difference. He points to the gemara’s (Berachot 34a) derivation from Moshe’s prayer for Miriam that one does not have to mention the relevant person’s name at all. While the Magen Avraham (see Mishna Berura 119:2) limits this to cases when the prayer is in the subject’s presence, we still see that an exact name formula is not crucial for efficacy. Therefore, if one does not know the mother’s name or there is another reason not to use it, the father’s name is fine.
Regarding many halachot and as part of the philosophy of conversion, the convert is no longer linked to his biological parents (see Yevamot 97b). Therefore, we would not use your friend’s biological mother for this identification. Perhaps you were thinking of using Sarah Imeinu, as indeed she was also a leader in the field of conversion, at least regarding women (see Bereishit Rabba 39:14) as well as a matriarch for all Jews, which might be important regarding one without a halachically recognized mother.
However, Avraham and Sarah are probably not of the same ilk in our context. There is a machloket whether converts can make the declaration of bikkurim, which includes the phrase "the land that you gave to our fathers." In explaining the opinion that he can (which we accept – Rambam, Bikkurim 4:3), the Yerushalmi (Bikkurim 1:4) cites Hashem’s proclamation to Avraham: "… for I have made you the father of a multitude of nations," which is brought as the source for calling a convert "ben Avraham (Avinu)" (Mishna Berura 139:11). While Sarah was an important spiritual mentor in her time and is a matriarch of Bnei Yisrael, we do not have sources of this magnitude regarding being a mother figure for faith seekers from all nations.
Therefore, it would seem that your friend told you her name correctly. As far as whether to add in the word Avinu (to distinguish from the many Avrahams who live in our times), when the name’s use is of halachic significance (e.g., a get), Avraham Avinu is used (Shulchan Aruch, Even Haezer 129:20). Regarding aliyot, where the name is less crucial, many use just Avraham to not embarrass the convert or put his status in the spotlight (see possible hint in Rama, OC 139:3). We have seen above that exactness in the name is not very important for prayers (Hashem knows who is intended), and the convert can do it however she wants.
Lessons
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