Ah, yes. That thing.
My interpretation is that it is a generic doorway/room control panel, as we see it in more places than near a turbolift.
We see it on the bridge of course, in both turbolift alcoves near the door:
We also see the structure of one near the turbolift doorway in the shuttlebay:
So one may assume it's a panel for calling a turbolift. BUT....
We later see it in the officer's lounge in TFF near the doorway. Bones actually uses it, pressing the topmost button to turn on lights to disrupt Spock's "staring out the window at the stars in contemplation" moment:
The fact that it's here, not near a turbolift, suggets it was meant to be a generic door/room control.
We see it in TFF yet again in the brig:

This is the most interesting depiction. The top two buttons are omitted and the word SECURED in their place. Also, we can see that the bottom portion says "SECURITY SCAN". This leads me to believe that the lower portion of these displays is in fact a security scan feature of some sort, most likely a fingerprint/handprint scanner to access certain areas, engage security lockouts, and...obviously...lower and raise the brig forcefeild

This is evident by the placement of the five brackets arranged in a hand/digit fashion (what about crewmembers with more that 5 digits on a hand??

)
For the next film, I believe we only see them on the bridge:
And then in GEN, we see them again near the turbolift, the "fingerprint brackets" converted into five skewed capsule graphics, which could still be fingerprint scanners:
So, in summation, I think they're meant to be multipurpose control panels for things like generic room functions (such as controlling illumination), displaying readouts of turbolift status/availability, and security scans. Hell. Maybe even thermostat capabilities.
My replica, just one of the many color variations we saw of it: