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As background you should read here, a summary of grandparent rights in Ohio: http://www.lsc.state.oh.us/membersonly/127grandparents.pdf
Ohio does recognize grandparent visitation rights (not all states do). In your case, with the death of your son, you would have standing to request visitation. Generally, unless there is some major issue (like a history of violence toward kids or something) the courts will grant visitation to grandparents in these situations.
As for how much it will cost... that depends. I cannot tell you how much another lawyer would charge you or how long the issue would drag out. Generally, though, if it is contested by the mother, you are going to spend at least $1000 to retain a lawyer for a contested hearing.
Of course, you could file the paperwork yourself, and it would just cost you the filing fee, which is less than $200.
The botXXXXX XXXXXne is that you do have a right to see the child and be part of the child's life. I am sorry for your situation. But if you pursue the issue, you will most likely win out (barring some major omission on your part).