This the easy way, but please note you lose the entire Navigation bar. The Nav bar is more than just the Next blog link. It also has the search engine, but you can add another easily (just add a gadget). Plus it has the link back to the Blogger dashboard, and the Sign in and Sign out in the right corner.
Select Layout from the left menu on the dashboard and click on Edit in the "Navbar" gadget.
In the pop-up window scroll down towards the bottom and select Off.
Or you can follow the directions in the video to remove the Nav bar.
Here is the text that you need to copy and paste into the HTML:
Thanks to Dave Lepisto for making the video.
You may also want to take a look at The Real Blogger Status for more ways to hide the Next blog link.