I want to start some non profit blog (example.org). What are the process of starting organizational blog and what are the benefits of doing this? Please help.

Benefits? Well you don't get any profits but I guess you could generate some interested and with good content, some reputation. Finding a cause or idea, getting a domain name, and writing up some posts would be the logical place to start. Maybe find some people to help you out.

Why do you want a blog? Is it to keep members informed? Recruit new members? Solicit donations?

Whatever reason, a blog takes time to manage. To be effective the writing - spelling, grammar, and formatting - must be very high quality. It also has to be updated regularly. If you go too long between posts, people will stop visiting your site.

For professional appearance and ease of use, I recommend WordPress. You should probably also consider registering a domain name. If you host your blog on WordPress you can point the domain to it so people will see your domain instead of your WordPress account url.

You asked about the benefits of blogging, a good blog can keep your organization in people's minds. Letting the general public know what you do can get some people interested. From this interest you could gain new members and benefactors.

PayPal has a "Donate" button that can be added to a website. There are several other ways to use a website to generate revenue.

The one thing NOT to do is put the website up then ignore it. You must add new posts at least every few days. If you can't think of anything to write, search the article databases for free content. Be sure to give credit, and a link, to the original writer (and/or the site where you found it).

Well this won't really help you out all that much in terms of comparison to a .com domain name. Anybody can register a .org even if they aren't a not for profit, although it is somewhat looked down upon to use it for commercial purposes.

Basically, find a good domain name, set up wordpress on it, find a theme, and start writing. Then search the forum for ways to promote it.

That depends on what you are referencing. If you wrote the blog posts, then it's the same as quoting yourself. Useful to point out that you're already an established expert in a field, but a bit conceited otherwise.

If others wrote the posts you are using, then you can reference the blog as publisher, but be sure to include the author's name (screen name is ok, but real name is vastly preferred). In this case, the blog takes the position of the magazine in which the article was published.

If you are using the blog as an example of your work, then list it as a portfolio item rather than a resource.

One thing i want to say is that, you can host your blog on wordpress or host on your own independent host buy from a host SP, and then buy a .org(IMO,for the noprofit purpose) domain and point it to you host, but the key is that you are a professional in one filed, and you are good at writting, produce good conten for the visitors!

I would argue that if you need to ask what the benefits are, then there aren't any.

If you are creating an organizational blog so that people within the organization can communicate and distribute information, then that is the benefit.

You really should be asking, "what problem am I trying to solve?"
Whether or not a blog is the solution is beside the point

I'm going to set up a new site for a nonprofit org, and I'm going to look through free Wordpress templates, since we don't have much of a budget right now.

I think that the benefits of having a blog will be invaluable. It's really the only sure way to get visitors returning for more info, and for building an online community around the cause.