There are so many "ifs" "thens" to this question ...
If a newer city can't figure out how to change their city name to tell me what they want, then they get polish on their Builder's Hut. I want them to grow as quickly as possible so they're more likely to provide value to me later in the game.
I figure lower-level players need gold coins more than they need Culture. If the BH is full, then they get a click on their Main Hall.
If a player has a specific request, then I'll honor it, insofar as I'm able to. If I can't FIND the BH, it doesn't get polished.
In general, under 5k points and they get BH or MH. After that, they get the highest Cultural building of which I'm aware for their race.
Obviously abandoned cities get whatever I happen to click on. I'm there for the coins.