That depends on what you're talking about. The cost would be what is spent too produce a product or service. The price is what the consumer would pay. Rate is interchangeable with price, but it seems to me it is used more with a service than a product.
I'd say the average rate
All are good, but I would use the average price is $6 a month or the average monthly price is $6.
Cost can be misleading. It doesn't specify whether it's the cost of providing the service or the cost to the purchaser although most people would get the idea immediately. The other two are accurate.
Well, grammatically, they are all correct.
I assume you want to know which one conveys your intent.
"Average rate," "average cost," and "average price" may mean different things to different people.
I would be more interested in the "typical cost for month," but it is not necessarily better than what you have written.
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