I don't think anyone would suggest changing the grid system or standard font sizes as a way to differentiate your bootstrapped site. I wouldn't mess with the scaffolding type stuff unless you really need to create a specific desired effect. Off the top of my head, the quickest ways to make a bstrap site look unique and keep the haters happy would be:

1- Navbar, the dark gray will not fly, tweak the height of it too or remove it all together.

2- Background of your container, give it some life. Here is a good place to start- http://subtlepatterns.com/

3- Special font for your branding/ H tags, boom (http://www.google.com/webfonts)

4- change up the button colors a tad, if only one, the blue one, it seems to be everywhere.

The above items will take you 15-30 min depending on how many combinations you try out & will take you pretty far.