UPDATE 4 December 2007: This trick, for reasons unknown, no longer works
New Blogger has introduced private blogs where you can set the blog to be viewable by those you invite or by blog authors. However, Blogger have left out one thing, that is to make provision for private blog owners to enable people to request for an invitation. I have found a way around that. The only thing is, you will have to accept some changes to the appearance of your private blog.
What you can do is to add to the title of the blog a message such as (email email@example.com for invitation) and any other information that you may want, example, amplify the message a bit by adding "Please give reasons why you want to view this blog" or something similar. The only thing to remember is if you give the email address as shown, it may be harvested by bots for spam. I have a email me link in this blog which can also be harvested for spam, but I don't care because the email address I provided in the mailto: HTML to display the email me link is a special gmail address made for such purpose, and my experience is all the spam messages go into the spam filter, and I do not get spam messages in my Inbox. If you are worried, perhaps you can write the email address as xyz at gmail dot com.
You can see an example in this test blog: Peter's Private Blog.
To change the title, click SETTINGS > BASIC in the Dashboard.