Free hosting providers earn on the ads revenue (receive some minimal profit), while users don't pay for the web hosting. There's no technical problem, but If you'll have built a website on a sub-domain, you'll be all the time displaying their domain name. Free web hostings will want you to promote their company. Again, it's other kind of an advertisement.
I'm not so sure how will react the customers watching at free hosting ads and weird domain name having no association with your family company. Maybe, you ought to join a pro web hosting. Actually, you've already created a topic about the best place to buy domain names
. I hope you'll go for it.
Here're two very simple to use:
But you can have create a free phpbb3 forum from rather of free hostings. They don't have to specialize for PHPbb3 forums.