I am glad this question exists, because now I get to talk about the first successful RPG character I ever created...
GESAN THE HALF-ORC MONK
Gesan came from a faraway land (not Asia) where he was taught to hone his physical strengths and the virtues of mediation and devotion to his country and cause.
He was also, despite frequently participating in meditation, as dumb as a sack of bricks. And to this day he is one of my favorite characters I have ever played.
Here is how he uniquely contributed to the party, and to the enjoyment of the rest of the group.
GESAN PROTECTS EVERYONE
Gesan was a monk, but he was also a big, strong guy who could take a ton of hits, due to several good hitpoint rolls and a high constitution modifier. As such, he would often throw himself in front of other people to protect them out of his devotion to the group.
Usually this meant Gesan needed a lot of healing afterwards because he had done so before considering that the person he was saving wore heavy armor and had even better capacity for tanking damage than himself.
But he did it anyway. He had an idealized vision of himself, and this was stronger and more pertinent to him than the actual person that he was. "Dumb" individuals act like this all of the time, because they don't so much think of themselves by the qualities that they have, but the qualities they aspire to have. It works for them (Or it hopefully does if you're playing a character this way) because by aspiring towards those qualities, they obtain them to some degree.
But they still act towards them, even when they don't fully measure up to those qualities in reality. This is why Gesan throws himself in front of a heavily-armored Paladin, and has to be re-directed constantly by the other members of the group towards better goals. You will know when you are doing this right when everyone in the group shouts your character's name at once.
Let them redirect you though, because while playing a bumbling overconfident character IS fun, it's only fun until it causes problems for the group. Unless your character is also very obstinant, you should let them give you a bit of direction first, THEN throw yourself at it with wreckless abandon.
GESAN BREAKS THROUGH DOORS
This is perhaps my favorite moment in Gesan's history. While travelling through a dungeon, our group came upon a locked door. After exploring a side-room, we found a key being guarded by a skeleton. Gesan picked up the key, and when we walked back to the room, Gesan was given instructions to open the door.
He proceeded to break it down with his fists. And we held onto that key for a good five session.
Obvious solutions, like opening a door with a key you just found, don't always come to a character who is 'dumb'. Think about a situation where you've tried to accomplish a task that was fairly simple, but because you didn't know how to do it, you wound up doing a lot more work than you had to, or wound up brute-forcing your way through it. This more difficult solution is a "simpler" solution, and is often the logic on which a 'dumb' character will act.
You can justify it in a few ways. Perhaps the character wants to show off a bit by doing things the 'hard' way, even though it's a very bad idea. Or perhaps they haven't connected the dots and they simply think they're being told to do the impossible.
Again, you should try to do this when it is less likely to cause a problem, and let the other players' characters talk you out of it if it WOULD cause a problem. You are playing a dumb character, but that does not mean you need to be dumb with your character.
GESAN THINKS VERY CAREFULLY
The last room in this dungeon had a very complex riddle, one that Gesan was certain he could solve given enough time to think about it. He sat down and thought about it, for quite. Some. Time. While the other characters solved the riddle easily.
And just as they were solving it, Gesan burst into the room, smiling brightly because he had finally figured out the answer! Which wasn't terribly good since the casters of the party were now fighting a released Chimera through two chokepoints. Gesan backed out very quickly.
There's a myth about playing 'dumb' characters that says they should never be allowed to solve complex or cerebral problems, because they're 'dumb'. While they certainly aren't the heavy-thinkers of the group, it can happen, either by luck, coincidence, or sheer determined effort. And it adds a level of depth to a character when you show that off. That they aren't 'just' the dumb character, but that they can occasionally think of things on their own, even if it's not always useful, not always on-time, or not always applicable.
More importantly, a dumb character will try. They will try as hard as they can to help with problems that they aren't necessarily good at, if tasked with it, or if they have to do it. And you can really make their personality shine by showing just how they deal with this adversity.
In Gesan's case, it was through sheer determination and a refusal to give up on the task at hand - the same way he deals with all of his problems. And the way your character deals with their own problems should reflect not just on their 'dumbness', but on the real personality behind it.
GESAN SUMMARIZES GESAN
"What he mean when he say Gesan is not smart, is Gesan is slow, but sharp like plow. Gesan takes time to learn new things, and Gesan think carefully on things, but Gesan get job done, and Gesan always try, even if Gesan not always do it right. Gesan try as hard as he can."