According to this story, Democratic legislators in Connecticut have drafted a bill to make it illegal for any party in Connecticut to be named “Independent Party.” If this proposed bill is ever signed into law, it seems likely that it would be held unconstitutional. The Independent Party of Connecticut has been very active starting in 2008. Currently it has ballot-qualified status for all statewide offices except President. It also has that status for U.S. House, 4th district; sixteen State Senate districts; and 39 State House districts.
There are ballot-qualified parties named “Independent Party”, besides Connecticut, in Delaware, Florida, and Oregon. In the past there have been such parties in Arkansas, Maryland, North Carolina, South Carolina, and Utah. In 2012 a U.S. District Court struck down a Tennessee law that banned the name “Independent Party”, but the Sixth Circuit reversed that only because of standing; none of the plaintiff political parties were named “Independent Party.” Thanks to Joshua Van Vranken for the link.