What do you think about putting a logo icon inside the business name? For example, a circular icon in place of an "O"?

I don't recommend placing the icon in the name. It reduces both the legibility of the name and the scalability of the logo.

Sure, it looks cool, but for a small business, the legibility/scalability are more important to the success of your brand. You don't want to make reading your company name any harder on your viewer.

The people looking at your logo are already in a hurry, and probably not paying a whole lot of attention. If you challenge them further by trying to be clever, you could confuse them, frustrate them, or even make them think that your name is something else – if they mis-read the icon as a different letter, then that's no good.

Placing the icon beside, above, or in some other orientation separate from your business name, then you'll get more impact for your logo.