Ahh interesting. Just out of curiosity, is the server just used for private work, research, or for teaching? https://rstudio.cloud would definitely be something to keep in mind if resource constraints are not an issue.
In any case, SSL is definitely the path you want to go down. So I would recommend setting up an nginx server on the same computer (listening on 80 and 443), proxying RStudio Server at 8787 or wherever you like, and only opening ports 80/443 in your firewall (depending on if you want to allow "upgrading" the protocol from http to https).
The way SSL works on a local network like that is you either need a Certificate Authority that says "yep, this is legit." Depending on the size of the local network, this may or may not exist. If it does not exist, then you probably will want to just use a self-signed certificate and then host the key on any computer that needs to connect securely. (In the absence of a Certificate Authority, the client has to have the key to be able to encrypt / connect securely).
I intend at some point to write up an article on SSL (hopefully I will get to it someday) 
Until that point, maybe these articles would be helpful?
Specific to Ubuntu / Nginx: