Is a larger beta weight a better predictor than a high t-statistic? Say there are two variables. Variable A has a higher $\beta$ weight than variable B but variable B has a higher t-statistic than variable A. Can either of these measurements, standardized $\beta$ or t-value, offer better predictive power over the other?  Must t-value and $\beta$ be taken into consideration together or not?
 A: It would help if you could answer @whuber's question.  However, in general, this question cannot be answered.  
The reason is that variables are almost always on incommensurate scales.  Consider the typical study that involves human subjects (as in medical research or the social and behavioral sciences), what covariates do we typically have?  It is quite common to include $\text{sex}$, $\text{age}$, $\text{weight}$, $\text{height}$, etc.  Now ask yourself: How many years makes you male?  How many centimeters equals a kilogram?  Etc.  These cannot be equated.  However, the ability of knowledge of one variable to help you differentiate amongst possible values of another variable is a function not only of the slope of the relationship between the two variables in question, but also of how spread out those values are1.  That is, you cannot elide the issue of the incommensurable units with which your variables are measured.  Thus, @StocSim's answer, while commonly believed, will not work: the significance of your variables (their p-values, t statistics, etc.) is related to how spread out those variables are in your dataset and there is no absolute2 way to determine whether the range of your $\text{heights}$ is comparable to the range of your $\text{ages}$.  
1 See my answer here: how-do-you-interpret-a-low-coefficient-yet-statistically-significant-with-a-high, for more along these lines.
2 It may be possible to equate variables by recourse to something else, e.g., how much it would cost to increase the amount of different variables.  Note, however, that this will change over time with changes in technology, the market, etc.  
A: $\beta$ and t-stat tell very different bits of information about a variable's effect in a statistical model and they are not exchangeable:


*

*$\beta$ (standardized effect size) is far more important. It tells how strong or meaningful the effect is. B (unstandardized effect size) is usually easier to interpret and so is preferred to explain what the results actually mean in real-world terms, but $\beta$ (standardized effect size) is more useful for comparing the relative strength of effects between various variables, since the B measures are on different scales.

*t-stat (statistical significance) is also important, but it only tells you how reliable the measure of $\beta$ is. That's all it does. It absolutely does not tell you how important a variable is, and although it is very, very widely misused in that way, that is a misuse. (The same, of course, applies not only to the t-stat, but to all measures of statistical significance.)


Taken together, the $\beta$ and t-stat give you the effect size and how reliable is the estimation of the effect size. There are four general possible scenarios:


*

*High $\beta$ and signficant t-stat: The high $\beta$ means that the variable is indeed important and the significant t means that you can trust that the high $\beta$ is reliable.

*High $\beta$ but insignficant t-stat: The high $\beta$ means that the variable is possibly important, but the insignificant t means that you cannot trust this result because your sample size is too small for you to be confident in the results.

*Low $\beta$ but signficant t-stat: The low $\beta$ means that the variable is not important and the significant t means that you can trust that the low $\beta$ is reliable.

*Low $\beta$ and insignficant t-stat: The low $\beta$ means that the variable is possibly not important, but the insignificant t means that you cannot trust this result because your sample size is too small for you to be confident in the results. However, since an unimportant effect of the variable would require a very large sample to register a significant t, if the sample is reasonably large and $\beta$ is quite low, it is reasonable to conclude that the variable probably has very little, if any, effect. 


In an ideal world, all t-stats should always be statistically significant; if you have a huge sample size, then this should always be the case. In practice, the harder it is to collect the data, the harder it is to get a large sample size. Thus, #4 (low $\beta$, insignficant t) is often interpreted as a low or non-existent effect. However, if the sample size is not sufficiently large, then #2 (high $\beta$, insignficant t) cannot be ruled out as a non-existent effect. It simply means that your sample size isn't large enough for you to be sure, but there might indeed be an effect.
A: You're getting at two distinct concepts:


*

*Statistical significance

*Material significance, some notion of practical significance.


In economics, people call number (2) economic significance. In biology, I imagine they call it biological significance.
Statistical significance can be seen from the p-value (assuming your standard errors etc... are done properly). Material significance, economic significance is a more field, problem specific question.


*

*Is a 0.05% difference in return materially important? Almost certainly no for high risk venture capital investing, but almost certainly yes for short term rates on near risk free securities. 

*Is aerodynamic drag requiring 5 watts of power at 25 mph materially important? Almost certainly not for a car, but quite possibly yes for a pro racing cyclist.


Does a big number imply material significance?


*

*Of course not because you can always make a number big by measuring in tiny units! 


What if you standardize a variable by its standard error, making the estimated coefficient at least invariant to units?


*

*This may be better, but as @Gung discusses, it still doesn't answer how this effect relates to other effects. Is the $\beta$ in question measuring the effect of a small brook flowing into the Niagra river while the massive Niagra falls is nearby? 


Answering the question of whether an effect is materially important is inherently field and problem specific. Generally speaking, you want to capture the materially important effects when forecasting.
A: Beta weight itself does not say much.You should also consider its standart deviation, therefore t statistic is better for you to understand which variable(a or b) has a greater effect.
