Related to a previous idea, take the tweets from last night and pick 50 of them at random. When you wake up, you'll have an email waiting for you with those 50 tweets. It would pick the tweets from where you left off if you have a $25/month Tweet Marker account (I might be interested in a per-account price if it was in the low single digits per month, say $2/month) account, or just from a certain time if you don't. If I thought for a second you could reliably embed tweets using Twitter's JS, I'd suggest making them all embedded tweets so that Cards show up too. Set the Tweet Marker to the latest tweet at the time when the email was sent. You'll just have to be OK with missing the tweets not included in the email, safe in the knowledge that a computer looked deep within your soul and chose tweets based on an algorithm you half-understand.
The bot/app would be called Tweets From Last Night.