I have a 12 week old Cairn Terrier. She is doing well with potty training, a few accidents in the house, but not many. The problem is that she starting peeing in her crate every night about 3 weeks ago. I've washed and washed the bedding and even replaced the bedding and the location of the crate as suggested by the vet. I'm thinking her crate may be too big. It is about 20"by 12" and she can freely move around in it. Should I replace it with a smaller one? Also it is a canvas one, maybe it is retaining the marker even though I scrub it down...

I don't have near the confidence in a smaller crate some people do. I have started many puppies in a 36'' crate with no problems. Of course none of them were your Cairn Terrier. With some puppies, a smaller crate might help. You might try blocking part of it off with something, even cardboard might do.

Here are some other things that might help:
Housebreaking starts before you get home with the new puppy. If you don't have
A crate, buy one. I prefer the more enclosed, den like plastic ones. Skip the
Bedding. At first it gets wet, and later it can be chewed into choking
Hazards. A wire rack in the bottom will help keep the puppy up out of
Accidents at first. They are available with the crates, but a piece of closely
Spaced wire closet shelving from a home supply place is cheaper. If you
Already have a metal crate, covering it may help. Just make sure you use
Something the puppy can't pull in and chew. Dogs that start in crates as
Little puppies, accept them very well. Never leave an unattended puppy loose
In the house. If nobody can watch it, put it in the crate. I suggest letting
The dog have its crate all its life.

She could be doing it in protest of being crated. It is important not to let her take over and run the house. A top dog is not a drill sergeant, but a more equal among equals. Having a good pack structure reduces such problems. The dogs see all the people and dogs in the household as a pack with each having their own rank in the pack and a top dog. Life is much easier if the 2 legged pack members
Outrank the 4 legged ones. You can learn to play the role of top dog by
Reading some books or going to a good obedience class. A good obedience class
Or book is about you being top dog, not about rewarding standard commands with
A treat. Start at http://www.dogsbestfriend.com/. If you do take such a young puppy to a class, make sure they require everybody to have proof shots. Even if she has had all her shots, she may not have developed the immunity she needs.

Starting so young, it is much easier to teach obedience. You must keep sessions very short, maybe only a few repetitions at a time. In a month or 2, you will have her trained using only spare moments you never missed. You will also establish a foundation of good behavior and the correct relationship with her.