How can I identify and choose instrumental variables? I am interested in what effect civilian funding of rebel groups (X) has on their degree of violence (Y). There might be an endogeneity problem due to omitted variable bias and reserve causality. Hence, I want to account for that using instrumental variable regression. 
I am having difficulties in identifying/choosing the "correct" instrumental variable? Are there any rules of thumb, general guidelines or methods how to decide on a suitable instrumental variable? Or, do you just decide on it through theoretical considerations or evidence found in past research? 
What shall I do if I can't (theoretically) identify any instrumental variable which would be suitable for the X in my model? And what if, everything that effects X is endogenous and simultaneously affects Y?
 A: You certainly can choose candidate instruments "through theoretical considerations or evidence found in past research".
Then a simple check is to compute their linear correlation with the suspected endogenous variable, and their linear correlation with the dependent variable. This is the one of the two needed properties of an instrument that we can measure. If linear correlation is weak (below, say, 0.5), then expect results of poor reliability, if your sample is not really large (more than a few thousand observations).
You then ask 

What shall I do if I can't (theoretically) identify any instrumental
  variable which would be suitable for the X in my model? And what if,
  everything that effects X is endogenous and simultaneously affects Y?

I sense a misunderstanding here: if by $Y$ you mean the variable designated as "dependent variable" in your model, then, as I wrote previously, the instrument must certainly be correlated with the dependent variable.  
Considering the basic IV estimator, with $Z$ the instrument matrix $X$ the original regressors we have
$$\hat \beta_{IV} = (Z'X)^{-1}Z'y$$
Think what will happen if $Z$ is orthogonal to $y$.
If you meant to ask "what if, for all good candidate instruments, a case can be made that they are also correlated with the unknown error term?", then you can do the following:
1) Run regressions with and without instruments and compare and discuss results. In any case, a shadow of doubt will hang over their reliability
2) Look into Copula modelling and a maximum likelihood estimation with a Copula that captures the possible dependence of regressors with the error term.
