NotFoundHttpException handler should take into account it's parameters

Seems reasonable to me. I would call this a minor DX improvement, not an API change. Refiling accordingly.

Given that this is just a small improvement I think this rather belongs ont 8.1

Disagree. This isn't a feature addition and its disruption should be practically zero. It should also be fairly simple I imagine.

Giving a $event->getMessage() to the 404 and 403 pages of ExceptionJsonSubscriber.

And set to "Needs review" obviously.

+++ b/core/lib/Drupal/Core/EventSubscriber/ExceptionJsonSubscriber.php
@@ -39,7 +39,7 @@ protected static function getPriority() {
+    $response = new JsonResponse($event->getMessage(), Response::HTTP_FORBIDDEN);

I believe you need $event->getException()->getMessage().

Hopefully did a better job this time.

Just doing that would be bad, because just the exception message would be not valid JSON. What about using ['message' => $exception->getMessage()]
It would be also great to return [] in case the message is empty.

When you have done that, we also have to adapt the existing tests, as they expect a different message.

Added message as array so as to create valid JSON. Tests to follow

Updated failing tests due to changed response.

Unlucky 13! Somehow missed the tests on the patch. Apologies.

It is fairly simple, it seems. :-)

This issue is a normal task so we need to outline how it fits within the allowable Drupal 8 beta criteria. Can someone add Drupal 8 beta phase evaluation template to the issue summary.

Adding table. Also reclassifying as a bug. Back to you, Alex.

Edit: We also would not need a change notice here as no APIs are changing.

I agree with the re-classification as a bug. Committed 64332fe and pushed to 8.0.x. Thanks!

Thanks for adding the beta evaluation to the issue summary.