First up, you'll want to know how to enable these. Keep in mind that enabling everything possible in Notification Center will likely give you carpal tunnel, so try to be selective. Let's start with the completely free apps. There's the Vice app, which lists top stories in the “Today” view. Also totally free are Yahoo! News Digest, which will deliver a morning and afternoon edition to you at the time of your choosing, and News 360, which gives three top stories (with images) on your lock screen.
News 360 showing headlines in Notification Center.
There are also news apps that have a paid component, such as the New York Times NYT Now app, which has some free content but gives you full access with a subscription fee of US$8 for four weeks (or free if you have home delivery). Other news apps with limited free access and unlimited access with subscriptions include the Wall Street Journal app (varying rates for subscriptions) and the app for the UK newspaper the Guardian (US$3.99 monthly).
These are just the ones I've discovered in looking for news apps with iOS 8 updates, so if you have a news app with a nifty lock screen widget, let me know!