Welcome to the Q&A Forum

That depends (naturally!).  What you want is for your blog and website to inherit domain authority (DA) and domain trust (DT) from each other.  It is a very common practice for websites to use subdomains for blogs because they use hosted blog services to power their blogs.  Google knows it a common practice.  From what I've read and seen with my own experience, Google will treat your blog and website as one site (by that I mean share the DA and DT)  if you: 1) have a small number of subdomains and 2) the navigation of both the website and blog are very similar. Frequently I have seen Google use the URLs from subdomains in sitelinks for the main domain, further supporting this conclusion.  Some more ideas to improve the domain - sub-domain connection. You could submit a sitemap to GWT that shows the blog in the subdomain for the site.  You could create a folder called mysite.com/blog that has a redirect to blog.mysite.com. (Last two ideas not validated, just ideas).

Hope that makes sense and helps.

I recommend using blog.mysite.com and pointing the A record for this sub-domain to a Wordpress or Joomla blog that you host at Hostgator or similar cheap but good hosting service.  Its pretty easy to setup and you have the benefit of your blog still being on your domain so you still build domain authority.

I did this for a client that has a Network Solutions store which offers a terrible blog (and SEO in general for the store, but thats a different story).  Her blog is in Joomla and hosted at hostgator, check it out here:  http://blog.grandexpression.com

Notice how the Home page button of the blog goes to the NS store, not the blog.