It was pretty close though. Single line templates need to be enclosed in quotes.
action:
- service: script.turn_on # lauch relevant script
data_template:
entity_id: "script.livingroom_{{ states('input_select.cycle_living_lights') }}"
The other way to do it is with a service template:
action:
- service_template: "script.livingroom_{{ states('input_select.cycle_living_lights') }}"
Because scripts are services. There is a caveat with calling a script like that though: Scripts - Home Assistant
Doing it with the service template will wait for the script to complete before moving to the next action (if there is one). Doing it with script.turn_on will run the script and the next action in parallel.