What do you mean a rebellious logo?
Well, rebellion in its simplest form is this: when everyone goes left, are you willing to go right? Rebellion means looking at the context your company is in and figuring out what is missing.
At the onset of every identity project there needs to be a common goal between the designer and the owner of the logo. This goal is assessed by drilling down on the story, voice and tone, character, ideal customer, and the mission, vision, and purpose of the brand.
After getting introspective about who you are, you need to see what's already out there, especially in terms of a logo. It's astounding to see how the marks used by your competitors are similar. In fact, I'd guarantee you find at least two recurring motifs like an image or color palette used multiple times. The goal here is ascertaining what NOT to do.
After observing what's already out there, now you can formulate concepts that are different from your competitors. My recommendation is to think of them in categories: a pictorial mark (something that is a representation of a real object), typographic (word-based mark), and abstract icons (shape-based marks that don't resemble something real). You'll find that at least one of these categories makes up the majority of your competitors' logos. Don't pick that one.
Black and White:
Every good mark works in both a solid black and a solid white version. If you have to rely on color to make the mark effective, go back to the drawing board.
If you're good to go on the black and white versions of your logo, move into choosing colors. Remember, the key is to choose elements that are different from what's already out there.
Logos are special because of the things they live on. By placing your mark on business cards, marketing collateral, signage, digital environments, and at varying scales, you can forecast against placement issues in the future. This saves thousands of dollars ensuring that the mark doesn't incur printing hazards down the line and can retain its structure regardless of where it goes.
Go get em rebels.
My workout wasn't that great this morning. I'm having a hard time writing this. It was difficult to read this morning. I kinda feel lethargic.
I could examine each of these issues and think of a solution to each one, but it would just be addressing the symptoms.
You see, the issue was that I got sub par sleep last night. While I could have some caffeine, get shot of taurine, or do anything that would give me more energy, it wouldn't be a long-term strategy. Fixing the root of the problem, my lack of sleep, is what will makes these symptoms less prevalent.
Branding works the same way. If your marketing efforts, design systems, naming conventions, and other efforts seem off, you might have to backtrack a bit. Most likely, whatever is wrong is a symptom of something deeper.
You can spend all the time you want cleaning up the garbage downstream, but you'd save a bunch of time if you cleaned up the filth at the headwaters.
You would not prescribe a cancer patient to use a band-aid as appropriate treatment.
Likewise, it'd be stupid to prescribe a logo to fix a broken brand.
You have to be willing to undergo massive overhaul to make massive change. Dive deep into the fundamental flaws of your startup. Things like being aimless, having no defined culture, no spirit, a lack of confidence or purpose. Once those are fixed, everything else becomes easier.
Don't think a band-aid will cure cancer.