I would like to see some improvements to the referral system that would help the referrer, the referee, and the site as a whole!
An alert when you have referred someone, as soon as they create the account. Then you can send them a welcoming message and offer help. At the moment, you only know if you regularly check the referrals page, and more referrals would stick around if this was implemented.
A notification to let the person know who referred them. Once they're on-site, they might need help finding their friend, so a note that says "You were referred by x, click here to drop them a comment" would be great.
Encouragement for pending referrals to come back and complete the process. For example, a quick message after a month of inactivity to remind them that the site exists, or the ability for the referrer to send a personal message as a one-time thing. This would be a great thing for all users, but particularly those pending referrals as at least one person should be invested in the idea of them returning.
More information about what pending referrals need to do to finish the process. Then if you're forcing your friends to check the place out, you can point them in the right direction.
The ability to add a username as a referrer after creating the account. Add some restrictions that it needs to be done within the first day or so to stop abuse. I've recommended subeta to some users who didn't know my username when creating the account, so I didn't get credit for the referral. It would take a lot of the frustration out of the process.

Yesssssss to everything. I also think there should be a mechanism - if there isn't already, at least - to add somebody's username as your referral when you sign up. Lots of my friends who signed up and tried it out didn't wait for the referral link from me, which was a little bit of a bummer, but that was several long years ago and I've burned all my bridges.