This may come across as ranty, or even a bit angry. It's not supposed to, especially as I'm not angry about it. This is aimed at Players. All players, not just a specific group of them. Why? Because everyone does this from time to time, even the best players. However, when you're sitting there wondering why you're bored out of your mind, wondering why your character isn't getting any scene time or action, take a second and ask yourself this: what am I doing to prompt what I want? Ready? Let's begin.
My Character Needs Impetus To Act
This is a common thing you see in a lot of games, especially games where the GM likes to keep some sort of control structure over the players. Maybe it is a military game where the players get missions from command, or a mercenaries game where they have an employer who pays the bills. Whatever it is, the character is designed to basically not do anything unless specifically prompted for action in some way.
Now, in and of itself this isn't a real big problem. When it becomes a problem is when the character, when finally prompted, takes the most efficient route to be back at a resting state. I'm sure you've seen it before if you've GMed. You ask the player what their character is up to and get a "do nothing" answer (i.e. I'm at the park.) You provide stimulus as the GM (you hear sirens and see a police cruiser racing down the street, a man is running away and trying to cut through the park.) The character then either ignores the situation, or solves it in such a way that it is over with no interaction or way for it to play out going forward. The character then goes back to a resting state and soon enough the player is bored, or at least looks it.
Why is this a problem? For one thing, it puts a lot of pressure on the GM. They have to come up with something for you again, but everything they come up with is going to be discarded, ignored, or torn through with maximum efficiency that the GM isn't even getting a chance to breathe. Add to this the fact that boredom can catch, and a bored player will often distract the rest of the table in some way, and it becomes a cascading problem that the GM has to handle.
My Character Wants To Be Left Alone
This is similar to the above issue, but also a whole lot worse at the same time. Why? Because with this the character is actively shutting themselves away. It is a passive-aggressive way of trying to get character time or scene time because with the character shut away and not reaching out the GM, and other players, have to work extra hard to get you involved in scenes and activities, and of course they will because they know you're there and need to play too. Only, whenever not prompted to do something specific, you shut the character away and expect others to go looking for them. You expect the plot to need your presence or for something to happen where if you're not there everything falls apart. Only, that's not fair to the GM nor is it fair to the other players. There is a reason almost every RPG I've ever read warns against playing the lone wolf or someone who will actively move away from working with teams at every possibility. Because it causes problems and makes things unfun for other people.
The Solution: ACT
The solution to both of these problems, and almost every other problem when you're bored at the gaming table, is to simply act. If the GM gives you something, run with it and turn it into something else. Don't just "solve" the situation as efficiently as possible like it is some sort of puzzle. Take it, own it, and make it your own. Follow the criminal back to his hideout and take everything in, or make them an informant. Turn the help into befriending the cop. Seduce the postal worker who brings you the package. Turn the situation into a scene and not just a prompt to be able to do stuff in, and you make the game more fun for everyone.
Just acting, and acting quickly, can do a huge amount for a game. Think to almost any show and the protagonists in it. How often do they just sit around and let things happen? Now how often do they, when things happen, keep working at it and poking it until it turns into something? If a building is on fire they don't just sit outside and wait for the firemen to go in and save someone, they act. Maybe that means running into the building to get people out. Maybe it means going and assaulting the fireman by the hose because he wants the house to burn down. Either way, you've acted and taken the scene into a new level.
Don't just take from the GM, give back. If the GM prompts you for something, prompt the GM right back with your action. Even if you want to keep it mundane, you can still do this. Even a simple "I'm going to the park, I'm looking for a child playing with geese" is better than "I'm going to the park." Heck, want to be less specific? "I want to go people watching in the park." The more detail, the more action you put in, the better the game can be. But you have to act.
This post is as much to me as anyone else. Just keep in mind whenever you are playing and feeling a bit bored: what can you do to make the scene more lively? Got an idea? Good, ACT on it.