A very strong correlation between the new feature and an existing feature is a fairly good sign that the new feature provides little new information. A low correlation between the new feature and existing features is likely preferable.
A strong linear correlation between the new feature and the predicted variable is an good sign that a new feature will be valuable, but the absence of a high correlation is not necessary a sign of a poor feature, because neural networks are not restricted to linear combinations of variables.
If the new feature was manually constructed from a combination of existing features, consider leaving it out. The beauty of neural networks is that little feature engineering and preprocessing is required -- features are instead learned by intermediate layers. Whenever possible, prefer learning features to engineering them.