Can anyone help me give a class a helpful name related to a design pattern?

var sut = new Tester().WithMugInCart().WithCreditCardPurchase().WithPurchaseSuccessful();

As you can see, you're building the process that the automation has to go through by chaining multiple methods to it. This could potentially be the Chain of Responsibility pattern (not to confuse with creation pattern the Builder Pattern). You're creating the "request" (which is the Tester class) and immediately run through a chain of instructions, thus gives you the flexibility to choose which automation test to run at a specific sequence.

Also, it seems like you're inheriting a characteristic from the builder pattern where each method returns the current instance which then allows you to continue and chain them into a specific sequence. 

Thanks - that's very interesting - I did wonder about the builder pattern but because it wasn't building and returning the object being tested I wasn't sure if it was legitimate - so with the Chain of Responsibility would a helpful suffix be "Handler" and the name something like "OrderProcessingTestHandler" and the method to run the test "Execute"?

so with the Chain of Responsibility would a helpful suffix be "Handler" and the name something like "OrderProcessingTestHandler" and the method to run the test "Execute"?

I'll give you my honest opinion here and tell you, no. Fancy names are just syntactic sugar, you're not forced to follow names exactly as the design pattern describes. Your method names are fine, more than enough. Don't let it worry you. Besides that, if you're worried about the names then you need to consider to refactor your implementation and follow the exact design as in the link I provided. Ending a method name with "TestHandler" wouldn't be right since it's not really a "handler", it's just a method. a handler is suppose to handle a specific event, so your correct approach would be to link multiple handlers in a specific sequence by having a method that accepts the "handler" method to it. Think of it like the builder pattern, you're "building" a set of "events" that occur at a specific sequence which when executed, they execute at the order they've been set.

var sut = new Tester().WithMugInCart().WithCreditCardPurchase().WithPurchaseSuccessful();

This is known as "fluent" design.