How do I choose an appropriate Instrument to solve the endogeneity issue? I have a quadratic regression model and one of the variables is endogenous. How do I choose an instrument to deal with this issue?
Generally, following Verbeek's guide to modern econometrics there are three main attributes a good instrument should have:

It should be relevant. The instrument should be able to explain your the variable being instrumented and through it also the explanatory variable. For example, in two stages least squares (TSLS) which is a model from the instrumental variable (IV) family the estimator converges to  $\beta_2 = \frac{cov(z_i, y_i)}{
cov(z_i, x_i)} $ where $y$ is the dependent variable, $x$ the independent variable that is endogenous and $z$ the instrument. If there is no covariance between $z_i$ and $x_i$ then the estimator is simply not defined.

Instrument should not be 'weak'. Generally it is not  enough that simply $cov(z,x)\neq 0$, instrument also has to be significant and generally the first stage of regression has to be significant itself. This can be tested by looking at $F$-statistics. A rule of thumb  for TSLS is that instrument is weak if the first stage $F$-statistics is less than $10$ (see mostly harmless econometrics by Angrist and Pischke).

Instrument should satisfy exclusion assumption. It should have effect on $y$ only through $x$ not in other ways. The instrument $z$ also has to be independent.


Moreover, more on a practical note if you are using a quadratic model the program will ask you also to provide an instrument for both the variable itself and  the quadratic term. As mentioned by Michael in his +1 comment you should check if the above assumptions still hold for the quadratic term as well.