Can we mention our own idea of a project in the letter to a professor during grad applications?
Sure, that shows proactiveness, interest, creativity and many good things. 

I'd suggest to publish it somewhere before, even if it's just a blog, arxiv.org, safecreative.org or your favorite option.

Otherwise it will be their idea, at least for some professors. Some of those may actually have had that idea (years) before you suggested it, so it may really be their idea. By publishing it you are able to prove that at least you got to the same idea/conclusion on your own, even if it was years after them. In case that matters to you.

Make sure to emphasize the research aspects of that project.