The short answer to your question is that the owner controls the plot, but there are some facts missing from your question:
How many plots are there? I assume there are at least three, that your grandfather is buried in one of them and that your grandmother intends to be buried next to your grandfather.
Why does your aunt object to a grave marker on her brother's plot? This is not a legal question, but could give some further insight into what is going on here.
Is this a public cemetery, or is in a private cemetery on land owned by your family? Many cemeteries in Ohio are run by local government such as villages or townships, and they often make up their own rules. Sometimes, old relationships and politics are involved, but this is less likely in a large metropolitan cemetery than in a small township cemetery.
Is your grandmother legally competent? If not, does she--or should she--have a guardian?
Most importantly, what is the evidence of ownership? A deed? Or something else? Or just family lore? Burial plot ownership records are often poor or non-existent.
Your father was buried in a plot, which may be enough to establish ownership. Cemeteries don't knowingly bury folks in plots they don't own, or against the wishes of whoever does. We've never researched this issue, but you don't want to spend money and energy on a lawyers if you don't have to. If this is a public cemetery, I would suggest you start with your funeral director, who may have some insight because these types of disputes happen from time to time. The funeral director may have a relationship with whoever runs the cemetery, and might accompany you to a meeting to discuss the situation. Some cemeteries have rules that all graves must be marked, so your grandmother's/aunt's instruction may be contrary to cemetery rules. And if the cemetery also owns a company that sells and installs markers, the person who runs the cemetery will have a vested interest in giving you the answer you want.
After you have spoken with the funeral director and the cemetery director, if you still need help, please feel free to contact me directly. It is outside our usual geographic client area, but I may be able to refer you to a colleague in the Akron area.Ask a similar question