Spencer Fry wrote a great post on "Why entrepreneurs should write." I would further add that the benefits of writing are so extraordinary that you should write a blog even if you have no readers (and regardless of whether you're an entrepreneur).
I have over 50 unfinished drafts. Some of them are just a few ideas scribbled down arguing with myself. Most of them will never be published, yet I got value out of writing all of them.
Writing makes you a better reader
Blogging has changed how I read other people's writing.
In struggling to find the right ways to structure and present my posts, I am much more attuned to what makes a good argument and what makes a bad argument. I am better at seeing holes in other people's reasoning.
At the same time, when reading I am less likely to fall into the trap of discrediting a post with weak counterclaims. In most any post, there are likely to be counterclaims that are based on exceptional cases. Internet commenters love to point these out. However, these exceptional cases miss the main thrust of the post, and by understanding the implicit backdrop behind a post's argument, I get a lot more value out of reading.
I'm also more aware of the style of good writers. I mentally take note of the ways good writers phrase their ideas. I'd always enjoyed Paul Graham's writing, but now I really appreciate how he organizes his posts. He has an awesome ability to suck you into his world and show you what it looks like from his perspective. I've learned a lot about good writing from reading Bradford Cross's blog; his posts have a clear arc and make excellent use of short paragraphs to keep the posts flowing.
Writing makes you smarter
Writing reveals holes in your thinking. When your ideas are written and looking back at you, they're a lot less convincing than when they're just in your head. Writing forces you to mature your ideas by thinking through counterarguments.
Writing helps you organize your thoughts in a coherent way. This makes you a much better conversationalist when these topics come up. I can't count the number of times I've had deeper conversations with people because I had matured my ideas offline.
Consider anything else a side benefit
Everything else writing gives you -- personal branding, networking, inbound opportunities -- are just side benefits. They're potentially very large side benefits, but they are not the main reason you should write.
You should write because writing makes you a better person.