Besides the obvious - the exception stack trace - the more info you can get is better. So you should get all the system properties as well as environment variables. Also if your application have some settings, get all their values. Of course you should put all this info into your log file, I used System.out her for simplicity:
System.out.println("----Java System Properties----");
System.getProperties().list(System.out);
System.out.println("----System Environment Variables----");
Map<String, String> env = System.getenv();
Set<String> keys = env.keySet();
for (String key : keys) {
System.out.println(key + "=" + env.get(key));
}
For most cases this will be "too much" information, but for most cases the stack trace will be enough. Once you will get a tough issue you will be happy that you have all that "extra" information