Statistical significance versus sample size Statistical significance of a (variable in a) model grows with sample size. Citing Gilbert (1986):
If one uses test statistics with constant size (i.e. a constant degree of confidence), almost any simple model will be rejected given a sufficiently large data set. 
...[On the other hand] any hypothesis can be maintained by testing it on a sufficiently short time series.
From a practitioner's perspective, this is troubling. Statistical significance is often used in convincing that a (variable in a) model is relevant; but once you know it grows with sample size, the argument loses much of its power.
Here is a related discussion (e.g. the answer by Mark Claesen).
Questions: 
(1) How to overcome the sample-size-dependence of statistical significance? 
(2) Can we adjust statistical significance to make it independent of sample size? 
(3) What to use instead of statistical significance? 
The goal is to find out whether a given (variable in a) model is relevant and useful.
 A: This is why I am strong proponent of confidence intervals over statistical significance! A confidence interval tells you everything that a significance result tells you and more. 
If I have a result where the 95% confidence interval for a mean is (0.1,0.2) I know that this is significantly different from 0 at the 95% level but I also know that it's still a pretty small number. 
To answer your questions:
(1) How to overcome the sample-size-dependence of statistical significance? 
Use confidence intervals!
(2) Can we adjust statistical significance to make it independent of sample size? 
We shouldn't! Statistical significance attempts to answer the question: is X different from Y? If we have very strong evidence that X is approximately 0.1, well, it's different from 0.0. 
When all you have is a hammer everything looks like a nail. If you do not care about this particular question (nail) don't use the statistical significance tool (hammer). 
(3) What to use instead of statistical significance? 
Use confidence intervals!
A: In the standard null hypothesis significance testing (NHST) approach, statistical significance and sample size are related to two other parameters - effect size and statistical power (Cohen, 1988). These four parameters form a closed-circuit system, where, usually, one of the parameters (either sample size, or effect size) is determined via three others.
However, some researchers argue that the NHST approach is fundamentally flawed and, therefore, they advocate a different approach, based on confidence intervals, as @TrynnaDoStat mentioned. This new approach is appropriately named The New Statistics. The introduction to that approach and guidelines for switching from NHST to the new statistics can be found in this research article.
The new statistics approach is described in much more detail in the book "Understanding the new statistics: Effect sizes, confidence intervals and meta-analysis", written by the statistical reform pioneer Geoff Cumming (this book on Amazon: http://www.amazon.com/dp/041587968X).
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Earlbaum Associates.
