You could do this with a load balancer, or you could randomize the list like @TimRice suggests.
I like this idea, but if all sites have a relay with the same short name, then you could probably just use that in the primary relay selection list instead of the failover, except in sites with multiple relays.
You could actually give a single relay as many CNAMES as you want. You could have a client select between many relays, but it actually only resolve to a single one. You could also have those CNAMES resolve to different relays later on. You wouldn’t want this to change all the time, but occasionally it would be fine as long as a particular client tends to get a particular relay except when a major shift happens in the number of total relays.
Also, read this: