Under an earlier version of knoppix (3.4, a derivant of Debian) I was able
to compile a webcam driver without errors.

Now under knoppix 3.8.1 liveCD, I had errors and
discovered that knoppix 3.8.1 does not contain the
necessary header files in the directory
/lib/modules/2.6.11/build/include/linux (The
corresponding directory did indeed contain the
required header files in 3.4)

So I downloaded the kernel source for 3.8.1 and
untarred it into /usr/src/2.6.11, which is where the
symbolic link /lib/modules/2.6.11/build points to.
(This is only possible now with UNIONFS, which is the
only reason I'm upgrading to 3.8)

I was hoping (guessing) that this would fix the
problem of missing header files, however now I am
getting a slew of error messages (from
/lib/modules/2.6.11/build/include/linux/kernel.h)
like:
error: parse error before "size_t"
warning: function declaration isn't a prototype
warning: conflicting types for built-in function
`snprintf'

Apparently this liveCD is not completely configured
for compiling certain source code, since these errors
did not come up in v3.4.

What do I need to do to prepare the 3.8.1 environment
for compiling?

Umm,Im not sure I understand...you want to compile the new knoppix liveCD,or the webcam driver under the new knoppix 3.8?