I ran my own blog from November 2006 till October 2014. All posts are still online, but I don't have time to update it anymore. Please note that all images and media files have been removed when the backup was moved to a new host in early 2016. Enjoy!
A lot of people now have a blog, many more are planning to get one and some people even run an entire website. An own domain name would be cool - but now that domain names are quite inexpensive it becomes in fact very hard to find an available one.
To purchase a domain name is not that complicated any more, there are tons of sites offering domain name services, and the most popular might be godaddy.com. The registration process is now quite simple in most cases, and connecting your domain to your website has become quite easy too. In most cases the domain name registrars allow using their nameservers or even offer complete hosting service packages. Blogging services such as blogger.com offer features to easily display the blog under your own domain name. Most registrars also allow to purchase a domain name under various country top level domains.
So purchasing a domain name is not the problem any more - today the biggest problem might be the availability of the domain, e. g. to find an appropriate name. Millions of domain names have already been registered, and probably all words you'll find in a dictionary aren't available any more. Many of these domains represent valid website, but a large number of them have also been registered by domain name traders who hope to get a large number of money by selling them later on (apparently the most expensive public sale of an Internet domain name to date is porn.com which was sold in 2007 for $9. 5 million cash).
Most combinations of words have also been registered, which only leaves two options: either you'll try to buy a registered domain name (from sedo.com for example), but then you'll have to spend a lot more for your domain name - maybe hundreds or even thousands of dollars for a single domain, which is not really an option for a personal blog, website, or even for a small company site.
As you probably won't opt for very long domain names (which reduces the chances someone will ever type in your name), you'll have to become creative. Just take a look at some popular web 2. 0 platforms: del. icio. us, reddit.com, bebo.com, flickr.com or meebo.com for example. You'll note that there's now a growing number of websites using intentionally misspelled (or, let's say, creatively transformed) names. The advantage of such domain names is that they're short, they can be rapidly typed on the keyboard and they look funny, which may even allow to remember them more easily.
So how do you find such a domain name? Well, I don't really know - open your browser, surf the web, try to find short words that look catchy and think of how they could be transformed in a way to invent some kind of a new word. Then check if no other company is using this (google it) and if it's still available.
It's still possible to find a lot of acceptable domain names this way. Recenty I launched the joopita.com web directory, and an online writing site called tapyr.com. These are just examples, and as you can see tapyr.com is even a 5 character domain name which is quite hard to find these days. Finally all you'll need is some inspiration and some luck.