Dear friend,
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.
Answered by: Rabbi Alana Suskin