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.
Let's start with 1's cape. Its tattered edges imply that it was scavenged from the ruins rather than made for him, and it has a very distinct red color. The only other pieces of cloth that color in the film are the red flags with black and white symbols that we see strewn across the city where it takes place. These flags are later shown to have been the banners of a fascist dictator called the Chancellor, the man who first ordered the Scientist to create the Fabrication Machine and then corrupted it. In short, 1 might just be wearing what amounts to a Nazi flag.
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.
Next, there's 1 hat. This garment's most notable feature is the copper one-cent coin that he has fastened to it. The first thought is that 1 wears this coin on his head because it happens to have his name embossed on one side of it. This may be true, but later in the film, that same coin is used to cover the face of a dead stitchpunk during a funeral. This suggests that the stitchpunks have some sort of belief about the afterlife and that they associate coins with death. Whether or not 1 shares these views, he could be wearing one of those death symbols as a way to intimidate the other characters into following his rule. He's reminding them without words that death is always nearby and that he alone has control over it.
Much like his cape though, this decoration seems to delude the leader in a very unfortunate way. By wearing a symbol of death on his hat, 1 literally has death hanging over his head all the time. That's not a good position for someone as insecure as 1 to be in. It may have fed his fear of the machines to the point that he became paranoid, which then pushed him to more drastic methods of keeping the stitchpunks in line. Other factors played larger roles in this of course, such as three of the stitchpunks deserting the cathedral and another one becoming more and more curious about the outside world, but that lingering dread in the back of 1's mind from his coin could have sent him on a subtle but steady decline over the years.
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.