Facebook – How to Ignore Specific Friend Published Messages in News Feed

It does not seem to be possible. There is chatter in the Facebook Support Community but nothing to suggest that it is a feature right now.

There was a time where you could at least have priority of the filters by dragging but that seems to have been removed in the redesign.

For now until Facebook brings it back , you are going to be stuck with a add-on like Feed Filter or just stick those people you know use mafia wars into a Friend List .

Facebook keeps on performing some sort of A/B testing, where few users (based on some business logic of facebook) will see one UI, while rest will see other UI.
I don't know how it could be possible at the time this question was posted, but currently you can give a try to these steps:

1. On sidebar of the homepage, you will see 'triple dot' next to News Feed link.
2. Click on this. If you don't have this option, it means right now this feature is not available to you.
3. You will see multiple options here, like:
   i. Prioritize who to see first: Select names of friends whose posts you want to see on the top of your timeline.
   ii. Unfollow people to hide their post: You can unfollow your friends right from here, so that their post will never be displayed on your TL, unless you follow them again.
   iii. Reconnect with people you have unfollowed: If you had previously unfollowed someone, you can refollow them right from here.
   iv. Discover Pages that match your interest: Relevant pages based on your activity will be suggested by facebook. Click like to get their notification.
   v. See more options: If you have any hidden apps then you can manage them from here.
   
   Please have a look at the screenshot: