That's how you convey the 'some' part when referring to edible and drinkable things in Russian - by using the genitive instead of the accusative. The 'some' bit is implied in this case: "Её дети хотят (немного/стакан) сока".
If you were to say "Её дети хотят сок", it would mean something similar to 'Her children want juice', possibly contrasting said juice with the lumpy milk that the woman in question was about to force-feed her children with.
Generally, if you're not making a point out of picking a specific dish or beverage (like you would in a restaurant, for instance), you'll want to use the genitive instead of the accusative in most circumstances.
Awesome response. I understood the sentence correctly, but I like how you just explained what it was saying without the ending, and how that specifically worked in a restaurant setting. These little tidbits are so helpful in grasping the larger picture. This is where I really love how duolingo can work.
As a side note: I often take notes on the discussions. When a point like this arises, I usually put "NB" ("Nota Bene" - Note Well) before the note, to remind myself that it's an important point. I've done this with French, Spanish, Italian, and German - and English.
Learning Russian is the first time in all my years that I find myself using NB.NB., to make "Note Well" remarks on my "Note Well" remarks. And also NB2, NB3, etc.
Learning Russian is a Bear. A big, tough Bear.
in English, at restaurants I've heard people say, "I would like the fish" - referring to a specific dish on the menu. That would be in the accusative rather than the genitive, correct? Whereas if you were trying to make up your mind and said to the waiter, "I'm in the mood for fish", you'd use the genitive?
@Jeffrey855877 Regarding the 'the fish' part, you're quite correct. Come to think of it, the restaurant setting would be one of those instances where not using the article would actually be analogous to the usage of the genitive case in Russian.
Thing is (and I really should have expanded on this in my original comment), the implied 'some' part that is omitted before the genitive noun in Russian doesn't necessarily refer to an indeterminate quantity. It might also imply "какого-нибудь" ('some kind of'), conveying that you're either not sure what kind of juice/fish you'd want to have served or you just don't really care about the particularities. Most of the times, the omitted part implies both (and it could go beyond even that, but that's outside the scope of the sentence at hand).
So yes, just like you'd use 'the fish' as a shorthand for something like 'filet de poisson façon meunière', you'd also use the accusative "рыбу" to indicate that you're making your choice carefully and knowingly. And just like in English, if there's a possibility of confusion (say, if there is a choice of fish dishes on the menu and you haven't agreed upon a specific one in your conversation with the waiter), the waiter will be obliged to ask you to clarify your selection.
It's clear from your comment that accusative would be used for "the fish" (a reference to a specific dish), and genitive for some vague general notion of fish - and genitive would also be used where there's some confusion about which of several specific fish dishes, i.e., it's not just fish in general, but it's not a specific dish either - sort of in-between. That flows natural from a menu, anyway, because there's only a limited number of dishes. It would be like going into a bar and saying, "I'll have some liquor." Genitive for "some liquor".