"The basic issue is: Does Ohio require a Social Security Number to identify a beneficiary in a will?"
No state requires that beneficiary SSNs be included in the will. Indeed, no identification number is really needed at all. Moreover, given that wills become public documents once filed for probate, no SSNs or other identification numbers should be shown on the will. What's important is that the will is written in a manner so that it is clear who the beneficiaries are. In the case of Ohio,
Chapter 2107 of the Ohio Revised Code contains the requirements for wills. Some estate planning lawyers ask for this information routinely as it does simplify a number of matters later when the will is actually probated. In particular, the SSNs of beneficiaries are needed by the executor of most estates for tax purposes. But the SSN is not needed on the will and isn't really necessary until the person dies and the estate takes over.