Clothes make the man, and in 1's case, this is probably a bad thing. Because of his lofty status at the start of the film, he wears a very king-like hat and cape in his first few scenes. He's presumably worn this outfit for most of his time as the group's leader, so it's fitting that he loses this attire piece by piece as his leadership fails over the course of the plot. There's symbolism to be found in his garments, and since 1 is quite the poetic type, he may have subconsciously chosen to wear them because of that symbolism.
It's been suggested in foreign dubs of the film that the Scientist modeled each stitchpunk after someone that he knew in life—with 1 being modeled after the Chancellor. Why he would model one of his creations after the man who caused the machines to turn against humanity is unclear, but since bringing the stitchpunks to life involved transferring a piece of his own soul into their bodies, the Scientist may have inadvertently given his memories of the Chancellor to 1. It's possible that 1 then "recalled" these memories, liked some of the Chancellor's ideas, and decided to govern his own people in a similar fashion.
The fact that he doesn't start to find his courage until after the coin falls from his hat further supports that the copper piece represents death. By losing that symbol, his fear of death probably becomes more abstract and somewhat easier to live with. Again, other factors play larger roles in 1's transformation after this point, but the loss of his morbid token and its false sense of power could have forced him to finally stand tall on his own.
I think what makes 1 so interesting to 9 fans is that we know he's one of the good guys. No matter how objectionable his behavior is, he's still acting in what he believes to be the group's best interest. The challenge then is to figure out his logic from the clues around us, which is very fitting in a movie about searching for the truth. We want to understand and like 1 because at the end of the day, we all know that the calloused cermudgeon has a soul inside.