I have lambda function defined sth like :
def lambda_handler(event, context):
#get constant json argument passed from cloudwatch event rule
...
This is based on NodeJS, but it should be the same for Python. The Constant under Input is a simple JSON encoded object, which can then be accessed using the event variable.
Input
{ "config": "uk" }
Lambda
console.log(event.config)
Found this entry as one of the top Google results, so hopefully this will help someone else.