I am using a Sony nex-5 camera with a manual Samyang 8mm fish-eye lens for panoramic photography.
The image stitching application (Panotour 2.3) does not recognize the Exif data.
How can I force my camera to include the Exif data for my lens?
You can't. The Samyang lens is all-manual with no electronics to identify itself to the camera. In theory, the camera could have some feature which would let you enter this manually, but since Sony doesn't sell any lenses like that, they don't have a strong incentive to add that ability. See, however What is the correct, standard EXIF metadata to add for Rokinon 8mm fisheye lenses? for suggestions for adding it to your photos in bulk after the fact.
It's actually not the camera you're trying to force to cough up EXIF information--it's the lens. The lens has no electronic contacts, and so cannot communicate any of the lens's EXIF information to the camera to record in the image's EXIF. If you were a Canon shooter, I'd tell you to go find an AF confirmation chip on eBay and glue it to the lens, but a search on eBay doesn't show any such chips for Sony NEX (e-mount), so I think you're out of luck, there.
The Lens Compensation app does not write to the EXIF of images (despite letting you enter focal length, etc.) and unlike prosumer Nikon and Fuji X bodies, (and just like Canon dSLRs) you can't manually enter information for manual lenses, and use the "preset" when the lens is mounted.
Your only option may be to log which lenses were used for which shots, and then to write lens-specific information to the EXIF after you've taken the pictures. I use Lightroom and the Lightroom LensTagger plugin to enter the EXIF information for my Samyang 7.5mm f/3.5 fisheye lens for micro four-thirds. But basically, all you have to do is overwrite the EXIF with some external means (e.g., ExifTool) before feeding it into your pano stitcher--or use a pano stitching package that allows you to enter the focal length information manually when it's not found in the EXIF (Hugin, PTGui, etc.)
Also, be aware that if you're attempting to shoot 360x180 full spherical panos with this lens, that you do need to be rotating around the no-parallax point correctly--particularly if you're shooting indoors--and that coverage cannot be done with a mere four-around images, unless you're using the mft 7.5mm version of that lens on an adapter, with the hood petals shaved. With the native E-mount 8mm version, you will probably have to do six around in yaw, possibly with a zenith and nadir for full coverage.
See also: How are virtual tour photos taken?