Probably not. With an H-1B visa, you are obligated to work just for the employer who petitioned for your visa. Working full time, part time, or even performing work as an independent contractor may be a violation of your obligation.This could result in losing your visa, deportation, and being banned from the United States.
What you’re referencing (setting up a service-based site that would be monetized) could be considered work. It wouldn’t matter whether you sent the money to a bank account outside of the United States or another person, you could still get into serious trouble.
To know exactly how setting up your website would affect your H-1B, talk with an immigration attorney. You should consider. We are a legal marketplace and we offer a great service that we refer to as Micro. With this service, you get to ask an experienced immigration attorney a question for a flat-rate (usually just $25). You’ll get a response to your question within 48 hours. This would be a great way for you to find out if there is a legal way that you can honor your H-1B obligations while realizing your dream of a monetized website.