First of all, let me explain what I think makes a good favicon. In my opinion there are 2 core aspects for a great favicon. These are:
Your favicon must be memorable
This is the most important thing a favicon is supposed to do. When someone bookmarks your site or sees your favicon in a tab among many other web sites you want to make them remember yours. Your favicon is the smallest bit of your brand so make it great.
Your favicon must reflect your website
In no way people will remember your favicon if it does not reflect your website. The easiest way to make your favicon reflect your website is to use your logo. This, of course, only works if your logo is good. Read: What makes a good logo? by Jacob Cass.
To create a favicon from your logo, photo or other memorable element of your website you can use a free service called favicon.cc. This is a great site with full of examples and good tools. My favicon was also created there. I just took the F from my logo, cut it out by an image editing program and uploaded to favicon.cc. The only alteration I had to make was adding more white balance to the letter so that it has more contrast.
Now you know how to make good favicon. What are your favourite favicons?