The brochure states that you may use "Visa, MasterCard, American Express, Discover Card, Diners Club, Japanese Credit Bureau or Disney's Visa Credit Card."
Cozumel, just like stores in the states, will vary from store to store as to what they will accept. We've used our Discover before, but have also had to use either a different card or cash depending on where we were shopping. If your friend only wants to carry one card along, I suggest a Visa or MasterCard. They are more widely accepted because the % fee the retailer has to pay is lower.
Hope this helps.