I actually find myself agreeing with the County Executive on this. "Oneida County-Griffiss International Airport" is too cumbersome. "Griffiss" has historical significance and already enjoys name recognition on its own among aviators (the the market for the facility) - recognition that does not need to be diluted with "Oneida County." Lastly, "Griffiss" signifies something bigger than "Rome," bigger than "Utica-Rome" and even bigger than "Oneida & Herkimer Counties" combined, because all of these are far to small to justify such a huge airport (and, more than likely, to keep it going financially).
A few years ago, some Syracuse-area legislators actually suggested that Griffiss be used as an alternative to their own Hancock International for air cargo, in order to avoid having to make some expensive additions and impacts to their neighborhoods. It's unknown if those additions were ever made, but if they were not, there's Griffiss' market: all of Central New York, from Auburn to Canajoharie including Greater Syracuse, Greater Utica, and Rome.
I once had a conversatiion with a commercial pilot who lamented the sad state of development at the former Griffiss AFB. From what I understand the facility is one of the premiere heavy-lift capable sites in the northeast and has easy access to highway, rail, and even canal transportation facilities.