Are there conventions in economics for coding the distance between states? I know there are lots of gravity models in economics that take as an input distance between countries. My understanding is distance is typically coded based on the distance between capital cities, either in $L_2$ norm or great-circle distance. Most countries are small relative to the distances between them, and so maybe it doesn't matter much how they code the center of these countries for purposes of trade. But in my application I'm looking at US states, and many states are large enough that the internal placement of the location for calculating distances between states makes a difference. Additionally, state capitals are often not the economic center of the state (Carson City, NV and Albany, NY come to mind) nor the geographic center (North Folk, CA is hours away from Sacramento, CA), so I don't know of obvious choices.
Are there any standard choices I should know about here?
As a resource to others here are some sample data:

Average Latitude and Longitude for US States
US Census Centers of Population
List of geographic centers of each U.S. state

Would anyone provide a list of the economic center of each state?
I think the answer depends on who is impacted by your measure of distance and for what purpose. Kennan and Walker (2011) Econometrica measure distance between states as "the great circle distance between population centroids" in an attempt to model moving costs. They also include an indicator for whether or not the state is adjacent.