First of all I would examine whether the model performs the task it is supposed to perform, i.e. account for the conditional heteroskedasticity in the data. That would amount to testing for remaining ARCH effects in the standardized model residuals by the Li-Mak test. If the model fails the test, there is evidence that it does not do its main task well.
Testing for autocorrelations of the standardized residuals, leverage effects and distributional goodness of fit as suggested by @Neeraj also makes sense.
However, be aware that a model that passes all test may be an overfitted model. That is, it describes the sample data well (actually, too well) but it is not likely to generalize successfully, e.g. it would not fit a new sample from the sample underlying population (or data generating process) well. Therefore, use of information criteria (which penalize overfitting) for selecting a model may be justified.
- Li, W. K., and T. K. Mak. "On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity." Journal of Time Series Analysis 15.6 (1994): 627-636.