There are several ways to do this. The simplest way is to export from EyeTV in the Apple TV preset. This gives a high quality h.264 file that can be edited.
With a little more work, you can get an even better result.
First you will need MPEG Streamclip (which is free) and the Apple QuickTime MPEG2 Playback Component (about $20, from Apple's online store.)
From your EyeTV recordings page, you should be able to right-click on one of your recordings and select "Reveal in Finder". This should bring up a Finder window of the EyeTV archive folder where you can see your recording with a .eyetv extension. It turns out that the .eyeTV file is really a container. Right-click on it and select "Show Package Contents." A finder window will open with the package contents. You will see a file with the extension .mpg. This is the MPEG2 file.
You can drag this MPEG2 file into MPEG Streamclip and export it into any editable format. I would recommend the Apple Intermediate Codec, with deinterlacing, at current dimensions, but you can experiment with the settings that work best for your purpose.
You can create a folder in your Movies/iMovie Events folder with the name for this Program or episode. Use this folder as your target location when exporting from MPEG Streamclip.
The next time you open iMovie, it will automatically generate thumbnails.
