It'll be amazing if we take it (저녁 먹을까요?) literally!!! We can EAT the night that is effected by the nature and time! Actually, it should be said = 저녁 밥 먹을까요? Shall we have a dinner? (or supper depends on the meal!) Here again, since the most Koreans eat rice, it contains the word '밥' (cooked rice) So, if you want to be really precise, '저녁 식사 할까요?' might be better. But saying '저녁 먹을까요?' is absolutely OK since it's understood as 'Shall we have a dinner?' eg: Instead of saying '머리카락 자르러 가요' (I'm going to have a hair cut), the most Koreans say '머리 자르러 자요' (I'm going to cut my head off). And yet amazingly, it's very normally spoken and accepted by Korean society!

- Should we eat dinner? => 저녁(을) 먹을까요? - What should we eat for dinner? => 저녁에 뭘 먹을까요? or 저녁으로 뭘 먹을까요? * 에 is like "at" and 으로 "as" or "for". - 아침에 뭐/뭘 먹었어? What did you eat at breakfast. - 후식으로 아이스크림을 먹자. Let's have ice cream for desert.

It means "shall we eat dinner"