An acoustic instrument can belly because of failure. Sometimes the braces split or come loose i.e. unglued from the top. Less commonly the bridge plate (flat plate of wood under the bridge) will split or come unglued. If there are failures, obviously you need to address those before worrying about the belly. Failures can be deceptive - an unglued brace may be pressed tightly in place by tension even though it's no longer attached or supporting the top. You can take the strings off and peek inside with a big dental mirror to see if there are obvious problems, and/or probe the glue lines on the bracing with a pick or the thinnest feeler gauge you can find. It might make sense to do this before taking it anywhere just to give you an idea for what you're in for.
ABGs can also belly because the particular instrument's design is marginal - if the bridgeplate is plywood, it's underbraced, etc. This isn't as simple to address as regluing a loose brace. You can add or replace bracing or bridge plates, but those are really major efforts and may exceed the value of the instrument.
There's a product called the Bridge Doctor which basically adds a cantilever under the bridge to counteract the rotation caused by string tension. On under-braced instruments it can be a life saver. If your instrument isn't damaged in any way, that's probably the easiest and most cost effective fix. Some instruments (i.e. Breedlove guitars) are built with such a cantilever from the start. You're adding mass which can change the tone, but it's a much less involved fix than "upgrading" the structural components.
