Trump said Martin is leaving his post as pardon attorney at the DOJ.
Democratic lawmakers are warning about potential conflicts of interest.
During an hourslong meeting that was conducted virtually, the board voted to add Trump's name to the building's facade so it would read "The John F. Kennedy Center for the Performing Arts Restored and Renovated By President Donald J. Trump."
The president will appear alongside Nassau County Executive Bruce Blakeman, who has he endorsed in the New York governor's race.
A firm whose former chief had misconduct allegations was given the work instead.
She will leave her role at the end of the month, Trump said.