Puppy settles on her mat

Use a food lure and encourage your pup to follow your hand to the mat. Reward your dog for going to the mat/bed. Increase your rate of food you put on the mat so your puppy learns that being on the mat predicts good things.
Next, begin to set up your cue. Like "go to mat" and reward your pup when he sits or downs. Eventually, "to to your mat" means go lie down. You notice I move the mat about and repeat the pattern. Dogs learn quicker with taught in patterns.
Then I began to add a bit more time between rewards. This just helps the pup learn to be in the mat for a bit longer. I quickly begin rewarding this pup for laying on the mat.

Your "stay" cue will be much easier with these foundation skills.