If the roots warrant a promotion, find a container that is at least an inch wider than the original pot (and don't shy away from funky). With the help of a drill, you can put in at least three drainage holes, which are key. Drill holes an inch or so apart using a metal bit. If you are working with a clay pot, use a masonry bit.
Caution: Drilling might damage a container. Don't try this with a valuable one.
Tease roots free from the root ball so they'll quickly penetrate the new soil. If you want to share the bounty with friends, separate off a "pup" with as many roots as possible. Slip the mother plant into its new container, adding soil around the original root network. Be sure to press the soil around the roots, eliminating any empty holes.
Water. Keep the roots moderately moist and position the plant out of baking sun for a brief adjustment period. One to two weeks should do it. Then place the plants in a sunny window. Viola! Your indestructible houseplant has received a makeover that will add life to your room for a long time.