The best resort you can stay at while trying to save money is at the Reef Resort on the west end...that is...if you want to get away from the crowds.
Other than that, there are some cheap hotels and resorts on 7 mile beach. I think the Comfort Suites is one of the cheapest, and a place decent place to stay if you're on a budget.
Go to Tripadvisor.com and check it out.
http://www.tripadvisor.com/Hotels-g147365-Grand_Cayman_Cayman_Islands-Hotels.html
I'd recommend finding a condo with a kitchen so that you can cook some of your meals. Also, if you snorkel, buy your snorkel mask and fins at Academy before you leave on your trip.