It depends. No. If you are writing a blog for your business or with a journalistic style, then generally: NO. Go ahead and put your name on there and take the credit you deserve. Since you're writing this blog to build your business and attract new customers, you want to tell everyone under the sun.
Yes. Do not put your real name on anything! If you don't fit the profile above, it is probably better to leave your name off the blog. No one really needs to know who you really are. In the long term, your life will end up easier.
Also, don't tell anyone you know about your blog including friends, family, co-workers, clients, employers, etc. The nice thing about a blog is the ability to speak your mind freely. If all of a sudden, your mom and your boss start reading your blog, you might run into some issues.
They might get pissed at you over something you wrote, or you are naturally going to hold back because you know they are watching.
You also would have to keep your politics to yourself if your name were out in the open. If you ever go out to get a new job, and your blog pops up in a background search (and it surely will), your political views might go against your proposed employer's.
So if you are going to blog, just keep it under wraps when it comes to your personal life. Everything will work out much better this way in the long run. And you'll enjoy the complete freedom to publish anything you want without fear of personal retribution.