Trust Relationship and Policies for AWS API Gateway and Lambdas

Your Policy for the lambda should set up everything your lambda is allowed to do. This includes passing a role.

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Sid": "Stmt1470153553000",
            "Effect": "Allow",
            "Action": [
                "dynamodb:*"
            ],
            "Resource": [
                "arn:aws:dynamodb:us-west-2:439753510372:table/YoYoDyne_Products"
            ]
        },
        {
            "Effect": "Allow",
            "Action": [
                "lambda:InvokeFunction"
            ],
            "Resource": [
                "*"
            ]
        },
        {
            "Effect": "Allow",
            "Action": [
                "logs:CreateLogGroup",
                "logs:CreateLogStream",
                "logs:PutLogEvents"
            ],
            "Resource": [
                "arn:aws:logs:*:*:*"
            ]
        },
        {
            "Sid": "Stmt1449789105000",
            "Effect": "Allow",
            "Action": [
                "iam:PassRole"
            ],
            "Resource": [
                "*"
            ]
        }
    ]
}

Your policy also needs to have a trust relationship.

{
   "Version": "2012-10-17",
   "Statement": [
       {
           "Sid": "",
           "Effect": "Allow",
           "Principal": {
               "Service": ["lambda.amazonaws.com", "apigateway.amazonaws.com"]
           },
           "Action": "sts:AssumeRole"
       }
   ]
}