Translation:What do they eat at noon?
I think everyone who is going to learn Chinese should know this statement below:
凌晨:3:00--6:00 早晨:6:00---8:00 上午:8:00--11:00 中午:11:00--13:00 下午:13:00--17:00 傍晚:17:00--19:00 晚上:19:00--23:00 深夜:23:00--3:00
You should know that when the word "noon" is actually indicates the word "afternoon" and so it should mean 中午 in Chinese and not 下午 which that is the time when we eat lunch (午餐)
Personally, I think the word BRUNCH will be more accurate than the word LUNCH at the time 中午.
That is not correct. If the question asks about a past event, it would have been
他们中午吃 了 什么？
The question can alternatively be asking about a past habit in certain context. In that case we translate it to
What were they eating for lunch?
(Edit 2018.02.25) "What did they eat for lunch?" is acceptable when the question is about a past action only but not about its consequences.
I was a bit hasty writing that. Often when we ask " What did they eat for lunch?" it is because something had happened (e.g. someone got food poisoning) or we suspect that they didn't eat or they didn't eat well. All these cases where we have concern about the consequence(s) of the action, we would use 了 to indicate a change of state - say, the eating action was done resulting in someone got poisoned.
Certainly that's not always the case. If someone asks this during a conversation about a past event and expects only a description of the action without concern on the consequences of the action, it is also applicable. You are right I had implied that 了 was mandatory, but that's not what I wanted to do. Thanks and I corrected the comment above by adding a footnote.
We don't use tense in Chinese. The time of reference is derived from the context.
So if the context is not clear yet between the speakers, we add more information in the sentence. You would be saying something like What did they eat at noon at such time ?, What do they eat at noon today ?