Not quite sure what happened here; Brandon over at EngineHosting re-ran the installation and now I see my info in RPM.
So for future New Relic users that want to integrate with Craft on EngineHosting - at its most basic, ask EH to get New Relic installed for you.
Exciting! Thanks @joshangell for the help. Sometimes knowing it should just work is the key. :)
Editing as requested. EngineHosting fully manages my site/server. I don't have root via SSH so asked them to get the New Relic PHP agent installated (the agent installation requires root). EH did install NR for me, but the first installation did not work and I do not know why. When EngineHosting re-ran the installation, the PHP agent worked and Craft (and other PHP apps, mostly the 'index.php' file for each) started reporting to APM.
That's not helpful for people on cPanel or Plesk hosts, admittedly, nor is it helpful for people with root access doing their own installations. But if you can get the New Relic PHP agent installed where Craft is, then it will report to APM. Looking forward to getting more specific metrics from Craft to New Relic down the line, but that is not likely to come from me :)
With that very basic installation I get information like this:
I hope that helps.