Can one make a Mi Shebeirach for an ill person if they do not know the mother's name?
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If they know the father's name they can say that instead (Yabia Omer 2:11).
If they do not know the father's name either, the Nishmas Avraham (vol. 4 42:13) writes in the name of Rabbi Elyashuv that they should say "bas Avraham" - see there at length.
I have personally heard "bas Sarah" being used rather than "bas Avraham".