When was his grandmother born in The Way to Rainy Mountain?

The simple answer to your question is 1880; however, that exact date is never given within the text.  We can extrapolate the answer to this question by considering a few important dates having to do with the Sun Dance. By doing this, we can figure out that Momaday's grandmother, Aho, must have been born in 1880.  Momaday tells us that his grandmother, Aho, was seven years old when the last complete Sun Dance was performed in 1887.  Aho was actually there to witness this particular Sun Dance with all of its sacred rituals completed.  This Sun Dance must have included the opening of the sacred Tai-Me bundle, the revealing of the Tai-me effigy, and the worship of that effigy on the pole in the Sun Dance lodge.  Further, Aho witnessed the very last, incomplete Sun Dance as well in 1890 when she was ten years old.  This Sun Dance was left incomplete because the soldiers from Fort Sill arrived in the middle of the ceremony.  The soldiers proceeded to scatter the Kiowa tribe before they were able to complete their sacred rituals.