On your actual question:
Should be straight forward.
There are also custom interfaces for dump1090-fa available, they might be useful:
The second one, i'm familiar with, you'd literally only have to add window.setInterval(followRandomPlane, 60000) in an appropriate place.
The function to select a random plane is already present.
I suppose you are asking about the local interface for the planefinder client.
I'd be surprised if they'd go to the trouble of implementing such a rather niche feature.
The webserver and local web interface is baked into the client, so you can't modify it yourself.