WASHINGTON — Douglas W. Elmendorf is an obscure figure beyond a narrow radius around Capitol Hill. As the director of the Congressional Budget Office, his nuts-and-bolts job is to serve as the official scorekeeper on the price of legislation and the referee on the budgetary and economic impacts of policy, from the Affordable Care Act to an increase in the minimum wage.

So it is one of the stranger surprises of the midterm election fallout that the question of whether to reappoint him to his post has become a hot topic of debate — among Republicans.

“Doug Elmendorf has been so exceptional, he seems to be trying as hard as possible to give a fair, unbiased review of academic literature on all these issues,” said Michael R. Strain, deputy director of economic policy studies at the conservative American Enterprise Institute, who called for Mr. Elmendorf’s reappointment in National Review. “That sometimes has greatly frustrated the administration and sometimes helped the administration. It sometimes has greatly frustrated Republicans and sometimes helped Republicans.”

Not so, say the most ardent conservatives, who want Mr. Elmendorf, a former Federal Reserve economist, replaced as soon as his term ends on Jan. 3. Grover G. Norquist, the small-government, low-tax activist at Americans for Tax Reform, has prominently laid out an indictment that includes putting a supposedly inaccurate price tag on the Affordable Care Act and Mr. Elmendorf’s “failed Keynesian economic analysis.”