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Actually, the Ohio statute (section 3515.03) makes it pretty clear that county boards MUST conduct a recount upon request:

"Upon the filing of an application, ... the board shall promptly fix the time, method, and the place at which the recount will be made, which time shall be not later than ten days after the day upon which such application is filed or such declaration is made."

As a practicing lawyer, I can tell you that "shall" ordinarily leaves no room for discretion.

For the full statute, see:

http://moritzlaw.osu.edu/electionlaw/procedures_recount_os.html#351503

Upon the filing of an application, or upon declaration by the board or secretary of state that the number of votes cast in any election for the declared winning nominee, candidate, question, or issue does not exceed the number of votes cast for the defeated nominee, candidate, question, or issue, by the margins set forth in section 3515.011 of the Revised Code, the board shall promptly fix the time,

SHALL applies to the margins set forth in 3515.011.... less than 1/2 of 1%