Will ANY prior knowledge assist in determining contents of a SHA1 hash?
It all depends on whether the rest of the input is guessable.

The best attack we know for finding the input to SHA1, given the output and partial knowledge about the input, is a brute-force attack:


For each candidate at the full input, try hashing that input with SHA1 and see if it matches the known output.


So, the running time of the attack will be proportional to the number of candidates we need to try.  If part of the input is known, the running time will depend on how easy it is to guess/predict the rest of the input.

For instance, if the unknown part of the input is a 4-digit PIN, and the rest of the input is all known, then the system is definitely vulnerable.  There are only $10^4 = 10000$ candidates for the full input, so you can try each one of them.  This won't take very long.

As another example, if the unknown part of the input is a 128-bit secret value, and you have no way of predicting that input, and all 128-bit values are equally likely, then the system is safe.  There are $2^{128}$ candidates for the input, and you can't try them all.  Also, they're all equally likely, so you can't even get a speed-up by trying the most likely candidates first -- all the candidates are equally likely.

In general, if the entropy of the secret part is at least 128 bits, you should be good.