I would like to name my daughter Ava. My grandfather's (living) name is Avraham but his nickname is Ava. Would it be okay to name my daughter Ava if it's a variation of grandpa's name? (Ashkenazi tradition).
[See also the following questions already on Jewish Values Online: http://www.jewishvaluesonline.org/question.php?id=59, http://www.jewishvaluesonline.org/question.php?id=184, http://www.jewishvaluesonline.org/question.php?id=840.
There are a number of reasons why this is not a major problem. One is that your grandfather's name is really Avraham, not Ava. Second, it is your grandfather, not your father. Third, you are not naming the child after your grandfather, you are (at least that is what you seem to suggest) just naming your child Ava because you like the name.
With all that, you might want to add a second name just so that it is clear your daughter's name is different from your grandfather's name, even though it already is.
Good luck, and may you ava lots of nahat from your daughter.
First of all, B'sha'ah Tovah ( may the birth be at a good time) - may your daughter be healthy and grow to a life of Torah, chuppah and maasim tovim (good deeds).
The essential answer to the quastion you ask turns outto be a simple, "Yes."
The explanation is a little longer. There is no limitation in the Torah or talmud on naming your child after a living person - in fact, as you may know from your clarification that you're following Ashkenazi tradition, Sephardim consider it an honor to name a child after a living relative, and their custom is to do so.
The two reasons most important for not naming after a living relative in the Ashkenazi custom are -primarily, in my observation - the superstitious one: that naming after a living relative may cause a child to be taken by the angel of death rather than the elderly relative. The less well-known one is that according to halacha (Jewish law), since one ought not to belittle one's parents by calling them by their first name, naming a child after that grandparent or parent might cause the person to be dishonored by casual use of their name by a younger relative.
To satisfy the first matter, one need only avoid giving the exact same name ("Ava," rather than "Abraham," should be fine). To satisfy the second might be more problematic. Technically, your grandfather's name is not Ava, but if it is generally the name by which he is called, you may want to reconsider whether this will be a problem for the second point.
In that case, you might want to find a slightly different name, such as "Avigail" or something similar, and simply let your grandfather know that you're naming after him. However, if you wish, you may certainly name your daughter "Ava" and not be concerned, as it is a perfectly valid tradition in the Jewish world to name after a living relative.
Mazal tov on the birth of your daughter. I think the only question here is how your grandfather will feel. If as I suspect he will be honored, I would go ahead. If on the other hand he is concerned that somehow it is like wishing him dead I would avoid it.
In this case, it is not a question of Jewish law but of psychology.
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