Part 1: Recent Notifications
Let's start by analyzing the current (iOS 8.3) offering for controlling notifications. Some default Apple apps get special treatment and have more options while some have less but this is what the typical app will offer you: Do you want notifications from this app? If so, how many do you want to be displayed in Notification Center? Will these notifications make a sound, appear on your lock screen and so on.
What iOS offers for notification management works great except for the "Show in Notification Center" option. Show the 5 most recent notifications should automatically remove old items once the notification count exceeds the set limit, instead iOS stores all of those notifications and displays them once you delete one of the recent items. Personally, I don't get why anyone would want to see month-old notifications that were hidden.
The solution? A new setting that allows you to control how long iOS holds on to notifications sent from a certain app. For example in Mail, you can specify that you only want Notification Center to hold on to email notifications for today, and once tomorrow starts, clear all old Mail notifications from that day. Most people don't read and reply to their emails in Notification Center so it makes sense to have the most recent emails for that day only and maybe reserve a permanent spot for emails from contacts you've set as VIPs until they're read.
Part 2: Expiring Notifications
There's nothing more frustrating than receiving a notification from a store's app for a deal or a coupon that is valid throughout the weekend but once that weekend ends, the notification just remains in Notification Center until you manually clear it. The fix is quite a simple idea, if that notification had an expiry date and could just remove itself once it served its purpose then there wouldn't be any confusion and Notification Center wouldn't fill up with useless notifications.
This works great for coupons but it works even better for live streaming apps like Twitch and especially Periscope, where live streams are usually very short. There have been countless times where I open Periscope through a notification to watch someone's live stream only to be presented with "this live stream has ended" because I was just a few minutes too late. Whereas if this notification was temporarily displayed for the duration of the live stream, then you're less likely to be disappointed and, no one wants a rapidly growing list of useless previous live stream notifications.
Other Ideas
Lastly, here are a few other things that could enhance the notifications experience that I came up with while writing this post. The first is a remind me later feature. It's an obvious feature that Apple already uses for OS X update notifications that prompt you to restart your computer to complete installation. These notifications usually have an "ask me later" menu with a few presets like in an hour or tomorrow. The second feature is also very obvious and already works seamlessly in the Messages app where a notification that is read on one of your devices is automatically removed from all of your other devices using iCloud. It would work perfectly for social media apps like Twitter and Instagram.