I understood that in simple (linear) regression, with low observation numbers, a critical minimum for the correlation value is used to validate a relation (Triola, M.F. 2004. Elementary Statistics). This value is higher when the number of cases is lower (and when the minimum p value for a significant result is smaller). However, this seems to refer to simple linear regression, so I imagine that for multiple linear regression the critical correlation value should increase even more. The only problem is that, so far, I have not found literatur on how much. Anyone knows this?
There is no critical minimum correlation to be valid. If the correlation coefficient is 0.03 one might wonder how useful it is even if significantly different from 0.
Validation with a small sample size: The second best approach is to run the experiment again using the same sample size. The best approach is to run the experiment again using an appropriate sample size. The existing experiment becomes preliminary results or a pilot study.
Remember that appropriate sample size tend to determine generalizability of a study. Therefore, to have appropriate external validity it will be important to run the experiment again with an appropriate sample size. In some instances the evidence may be strong but applies to an isolated case, as might occur when research deals with a small sample size or a unique demography.
What about no linear regression or no parametric regression? Or you can use the Bayesian inference approach.
So, I think these would give you a solution of this problem.
Yes - Timothy A Ebert has the right idea. Simulation studies often help us figure out how to proceed.
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