I'm not sure if anyone has examined this subject in a conclusive or comprehensive way, but the few examples of names either changing or staying the same during and after the Revolution tend to depend on local politics more than a centrally-planned national campaign to change names. Most of my examples will come from New York, which is the state I am most familiar with and can furnish some interesting examples.

The names that did not change were those around New York City. Loyalism had run strong in some quarters of the City and Long Island before the British invasion, and the British held Manhattan, Staten, and Long Islands throughout the war. Loyalists from across the Northeast fled to British lines throughout the war. While many evacuated with the British at the end of the war, some must have stayed. As such, there was not much of a push to rename Kings and Queens Counties, though Kings County is more commonly called Brooklyn today.

Further north, some names were changed. In 1782, the town of Hanover, settled largely by Palatine Germans but politically led by the loyalist-leaning Colden family, changed its name to Montgomery in honor of fallen Continental Army General Richard Montgomery. Interestingly, the militia and town committee (the revolutionary body that ran the town in the place of the deposed official government) were thought to be under "undue influence" of the Colden family, and thus were not trusted to arrest suspected loyalists in their own borders. Replacing the name Hanover, the German province where King George III's family was from, with the name of a martyr of the Revolution may have been a way to bone up on their Revolutionary credentials.

Further north, the lands around the Mohawk River were divided between Tryon and Charlotte counties, named after one of the royal governors of New York and George III's wife, respectively. This region saw some of the fiercest fighting of the war. Many of the inhabitants were Scots that had settled along the Mohawk after the Seven Years' War. When fighting broke out, the majority of these Scots, led by the powerful and influential Johnson clan that dominated the region, remained loyal to the King that had granted them their lands. These loyalists fought hard alongside their Iroquois (Native American) allies during the war, but the British surrender cost both the loyalists and the Iroquois their lands. After the war, the New York State legislature changed the names to Montgomery and Washington counties as Continental Army veterans moved into the region. Montgomery County later went through several subdivisions, one of which is named after Baron von Steuben, a Prussian officer who trained American soldiers during the war.