I need some suggestions on where to begin looking for a beach cabin in Galveston. Not sure which beach to go to either. Any ideas would be greatly appreciated! 

VRBO.com is a good place to see what's available. 

Lady .Beach Bum, here on KU, has one she rents

Her house is on the West side of the island across San Luis Pass.

If you are looking for something less crowded, then her's is the best. 

We loooove the West side of the Island. It is a drive East to get to things but also very quiet with less tourist. 

I own a place in Pointe West. There are many options for rentals there as well as condos. The cottages are nice and the large custom homes are nice. The condos have 2/2 2/3 and 3/3 options on the beach or the Bay side. As a renter, you will also have access to the beach club which has a pool and club with cabanas and the lazy river on the bay side. It is very family friendly. You can google Pointe West Galveston and see pictures. There is a private beach there where cars are not allowed to drive on the beach and chairs and umbrellas for rent. 

We rented a 2 bedroom plus "kid bedroom" 3 bath house in April for I think $150 a night. It was about 20 minutes from 60th street just on the other side of Jamacia beach. The two bedrooms had queen beds. The kid room downstairs had two sets of double/twin bunks a bath and a kitchenette. It included use of the golf cart, beach toys, towels, sheets, cleaning, etc. It's called Big Blue. We had a great time! 

VRBO is a great site to find vacation rentals anywhere.