Where to submit an incomplete work just for the sake of record
So is there any way that I can submit my incomplete work online, so that it gets recorded?


Congrats on your research idea! As I see it, you have two options:


Publish a blog post or put something up on a website.
Spend more time on polishing your work: ideally, by getting feedback from a professor or trusted expert. Within a few more months, put together the best write-up you can at this time. Then put the paper online, e.g. on arXiv.


Based on the information you provided, my preferred option in your situation would be (1). This is because:


You describe your idea as incomplete. If that is true, you don't want to present this as a finished paper to the community (attached to your name) when it is more of an idle curiosity that is not yet fleshed out.
It is very uncommon for undergraduates to be able to produce a publication-worthy paper without mentorship. So, what you need at this stage is a good professor, mentor, and advisor. Without that, statistically speaking you probably don't have the skills or knowledge yet to pursue this idea in the most effective direction.
If your brother is right that the idea is bad (I don't know your brother, so I can't say if he is just being mean :) ) then announcing it in a public venue like arXiv could be embarrassing later down the road. It doesn't really matter now (lots of crappy papers get published to arXiv), only later.


In contrast, writing a blog post has several advantages: for instance, it's not really embarrassing later (no one expects you to sound like an expert in a blog post).

Overall, I don't mean to discourage you -- the fact that you had a new idea and are excited about thinking about it is a sign that you would make a good researcher. Establishing a relationship with a professor is the natural next step, and they could also give better advice about whether this particular idea is worthwhile.