Load balancing sockets on a horizontally scaling WebSocket server?
First off, you need to define the bounds of the problem you're asking about. If you're truly talking about dynamic horizontal scaling where you spin up and down servers based on total load, then that's an even more involved problem than just figuring out where to route the latest incoming new socket connection.
To solve that problem, you have to have a way of "moving" a socket from one host to another so you can clear connections from a host that you want to spin down (I'm assuming here that true dynamic scaling goes both up and down). The usual way I've seen that done is by engaging a cooperating client where you tell the client to reconnect and when it reconnects it is load balanced onto a different server so you can clear off the one you wanted to spin down. If your client has auto-reconnect logic already (like socket.io does), you can just have the server close the connection and the client will automatically re-connect.
As for load balancing the incoming client connections, you have to decide what load metric you want to use. Ultimately, you need a score for each server process that tells you how "busy" you think it is so you can put new connections on the least busy server. A rudimentary score would just be number of current connections. If you have large numbers of connections per server process (tens of thousands) and there's no particular reason in your app that some might be lots more busy than others, then the law of large numbers probably averages out the load so you could get away with just how many connections each server has. If the use of connections is not that fair or even, then you may have to also factor in some sort of time moving average of the CPU load along with the total number of connections.
If you're going to load balance across multiple physical servers, then you will need a load balancer or proxy service that everyone connects to initially and that proxy can look at the metrics for all currently running servers in the pool and assign the connection to the one with the most lowest current score. That can either be done with a proxy scheme or (more scalable) via a redirect so the proxy gets out of the way after the initial assignment.
You could then also have a process that regularly examines your load score (however you decided to calculate it) on all the servers in the cluster and decides when to spin a new server up or when to spin one down or when things are too far out of balance on a given server and that server needs to be told to kick several connections off, forcing them to rebalance.
What I don't understand is how to effectively route new socket connections to the server you've picked with low socket count.
As described above, you either use a proxy scheme or a redirect scheme. At a slightly higher cost at connection time, I favor the redirect scheme because it's more scalable when running and creates fewer points of failure for an existing connection. All clients connect to your incoming connection gateway server which is responsible for knowing the current load score for each of the servers in the farm and based on that, it assigns an incoming connection to the host with the lowest score and this new connection is then redirected to reconnect to one of the specific servers in your farm.
I have also seen load balancing done purely by a custom DNS implementation. Client requests IP address for farm.somedomain.com and that custom DNS server gives them the IP address of the host it wants them assigned to. Each client that looks up the IP address for farm.somedomain.com may get a different IP address. You spin hosts up or down by adding or removing them from the custom DNS server and it is that custom DNS server that has to contain the logic for knowing the load balancing logic and the current load scores of all the running hosts.
Route the websocket requests to a load balancer that makes the decision about where to send the connections.
As an example, HAProxy has a leastconn method for long connections that picks the least recently used server with the lowest connection count.
The HAProxy backend server weightings can also be modified by external inputs, @jfriend00 detailed the technicalities of weighting in their answer.