There is no US capital named "for" a potato. The state of Idaho is used as the name of a type of large brown potato (Russet Burbank), which is grown there and elsewhere.
Vermont has a capital city named Montpelier.
The state capital that is named in honor of Thomas Jefferson is Jefferson City.
Vermont's capital, Montpelier, is named for a French city, Montpellier.
Idaho is the name of a US state, not a capital. Its capital is Boise.
Alabama's capital is Montgomery.
The capital of Missouri is Jefferson City, named after Thomas Jefferson, the third President of the United States.
Idaho.
Montpelier is the capital of Vermont.
Jefferson City, the capital city in Missouri, is named for Thomas Jefferson, the third president of the United States.
New Jersey
The state capital is Madison, named for James Madison.
Olympia is the capital city of the U.S. state of Washington.