I could probably write a novel about what NOT to bring on a dive boat but I'll try to keep it simple.

Somebody already illuded to a wet and a dry bag. It's definitely a good idea to have a separate dry bag with a change of clothing or your drysuit undergarments, extra contact lens/glasses.
There are a lot of variables but one of them is the size of the charter boat (e.g., small inflatable, smaller six-pack vessel, larger inspected vessel, etc). In general try not to bring large roller bags with wheels or huge toolboxes/bins onboard. Anything that's not collapsible / easily stowable especially on smaller vessels. You'll likely annoy the captain / crew.

There is always that one guy who has to bring EVERY piece of dive gear that he owns on a boat. I get it, just use common sense. There is definitely a limit to what extras and backup gear should be brought on a charter.
For local charters it's best to arrive with gear assembled (BC on tank with regs assembled). Much easier to do this on land than a rocking / swaying boat and of course it gives you a chance to verify everything which is much easier to do than on a cramped boat. Obviously this isn't always possible for vacation style charters.
A mesh bag works pretty well in the tropical or warmer water trips but a lot of local NY / NE divers use large milkcrates or 18-gallon rubbermaid bins to store their dive gear. You might find divers in your area do something similar.
There is fierce debate about which one is preferred

It seems NJ guys seem to like milkcrates but I tend to see rubbermaid bins more in New England. Some boats have a "system." A couple of the dive boats I use or worked on have benches where 18-gallon rubbermaid bins fit perfectly underneath.
They both have their own pros/cons. Milkcrates are smaller, take up less space and things can dry out more easily. The rubbermaid bins are larger, fit more gear and then can function as a bonus rinse tank for washing gear. In both examples I run a piece of paracord along the sides so I can clip things off inside the bin. Everything inside my bin has a bolt snap and gets clipped off or secured.
For a small RHIB / inflatable style boat I prefer not to bring any sort of bin if I can avoid it. In these situations I am already wearing my drysuit + undergarments and most of my "extras" will be stuffed in my pockets. Definitely try to avoid bringing any larger suitcases or bins onboard inflatable boats but this advice also applies to any smaller vessel where deck space and storage is at a premium.