See the main article (removing the attribution gadget from blogs with Designer templates) for the most recent information about techniques that do work and for background information about the topic
I have not checked the Blogger Terms and Conditions to find out whether you are allowed to remove the Attribution. Even if they don't mention it today, a requirement to keep the statement could be added tomorrow. This advice does not mean that I, or Google/Blogger, are saying that you should/can/may/must remove the attribution statement.
Note: the following instructions relate to the pre-Sept 2011 Blogger interface. They will not be updated, because this method (generally) does not work.
How to remove the Attribution Gadget by unlocking and then removing it.
1 Go to Design > Edit HTML.
2 Take a backup:
Download a copy of your template: this is a backup - save it somewhere safe, in case things go wrong.
3 Find this section in the template:
<!-- outside of the include in order to lock Attribution widgetThe simplest way to locate it is to use your browser's search function (Edit > Find in Firefox, similar commands in other tools), and look for "lock Attrib".
<b:section class='foot' id='footer-3' showaddelement='no'>
<b:widget id='Attribution1' locked='true' title='' type='Attribution'/>
locked = 'true'to
locked = 'false'
5 Preview the template change, to check that everything else is ok. If not, fix the problem. (The attribution will still be showing at this stage.)
6 Click Save Template
7 From the Design > Page Elements tab, edit the attribution gadget. It will now have a Remove button on the left hand side. Click this, to remove the gadget from your blog.
8 If you want to, add an an alternative attribution notice.
Removing the attribution gadget from blogs with Designer templates (current approach)
Giving your blog a credits statement
Types of templates in Blogger.
Template Designer now live in Blogger.
How the data in your Blogger blog is organised