The year 1935 was officially selected to represent Connecticut's 300th anniversary year by the Connecticut Tercentenary Commission in 1934. In arriving at its decision, the Commission considered multiple milestone dates in Connecticut's history that took place between 1633 and 1639.
Each of the following was considered as an establishing milestone:

1633 - the year that Windsor was first settled; originally Dorchester. It was Connecticut's first European settlement.

1634 - the year of Wethersfield's settlement; originally Watertowne.

1635 - the year Hartford was settled; originally called Newtowne.

1636 - the year of establishment of the Connecticut Colony when Windsor, Wethersfield and Hartford were united as a self-governing colony.

1638 - the year New Haven was settled - originally called Quinnipiac.

1639 - the year the Colony adopted Connecticut's Fundamental Orders. The Fundamental Orders framed the government of the Connecticut Colony and represented the first time in US history "when a popular government was established free from any power over and above the people themselves." The Orders served as a model for many State Constitutions as well as the US Constitution.
Note: All settlement references above represent European settlement activity; the area was home to indigenous people before the Europeans arrived.The Commission decided that 1935 would officially be known as the Tercentenary year, stating that "the beginning of settlement had been made" in that year via the river settlements of Windsor, Wethersfield and Hartford, plus the fort established at Saybrook. The decision set formal plans in motion for the State's tercentenary. This included the introduction of a commemorative coin bill in the House of Representatives by Francis Thomas Maloney (D-CT) in March 1934; the bill moved through Congress quickly and became Public Law 73-446 in June 1934.
Note: As a courtesy to its neighbor, Rhode Island, Connecticut decided against staging its 300th Anniversary celebrations 1936 to avoid conflicts with Rhode Island's planned 1936 Tercentenary celebrations. Nice state-to-state cooperation!1935 Connecticut Tercentenary Half Dollar

For other of my topics on commemorative coins and medals, including several on the Connecticut half dollar, see:
Commems Collection.