When you're stabilizing wood, you're essentially displacing the water and air pockets within the grain structure with a harder material. There are many methods to do this and some work better than others. Overall, they all exploit the porous sponge-like nature of wood to wick in a liquid that will harden into the air pockets.
There's a detailed article on the general procedure:
http://blog.woodturnerscatalog.com/2014/02/wood-stabilizing-v-2/
If you're looking for some kind of cheap, homemade alternative to expensive resins, you may want to try good old wood glue thinned out with water and vinegar. Also, you may want to try out thinned hide glue. Wood generally wicks in liquid through the end grain. Make sure you leave a nice freshly exposed end grain surface.
The following article discusses some alternatives for commercial resins:
http://dcknives.blogspot.com/p/wood-stabilization.html
He mentions that many are acrylics dissolved in acetone. You may want to also experiment with thinned clear acrylic paint or even thinned shellac.
There are several factors that effect the success, including the wood used. You're best bet is to experiment with different recipes.