I would have thought "the Death Star" would be "Hov Hegh" and "Hegh Hov" would be "the star dies". If that is not correct how would you say "the star dies"?
The verb Hegh cannot be used adjectivally. While it can sometimes be convenient to translate it into English as "to be dead", Klingons consider it less a verb of quality and more a verb of action: "to die".
If this were the verb Hegh, it would, indeed, translate as "The star dies." In this exercise we are not actually using the verb Hegh, but rather the identical looking noun Hegh. When two nouns appear together like this they might be followed by a conjunction to join them into a list (but that doesn't happen here), or they might be parts of separate clauses, only appearing together because the clauses are next to each other (that does not happen here and it's probably best style to use a comma in such cases), or the first may be acting in a genitive or possessive manner (which is the case here). So Hegh Hov, in this case, means, "Star of Death", "Death's Star", or "Death Star". Hov Hegh would most likely mean, "The death of a star", "a star death", or "a star's death".
When there is an identical verb and noun (or other parts of speech), it can sometimes cause confusion. Context often makes it very clear which interpretation you mean and can sometimes be strong enough to make you think only of the proper interpretation, even though in other contexts you might think of the other interpretation. The problem with Duolingo is that it usually gives you no, or at least very little, context. In this case, the lack of a period might have been a clue that this was not meant to be a complete sentence, but just some sort of phrase. None the less, if you had tried, "the star dies", it would have been accepted, since that is a possible interpretation of those Klingon words in that order.
Star of Death and Death's star both make sense, but it seems a leap to take that to "Death Star". But the clear precedent is "Hegh bey". Thank you for the very well thought out response.
But I am curious as to the source of death not being used as an adjective? That is interesting and I can't find a reference. How would you then say deathly ill?
Per boQwI', Hegh is both the noun for "death" and also the verb "to die" (a verb that can be used transitively, somehow). Consequently, it would seem that Hegh Hov can mean both "The star dies." and also "The star of death" (or, "Death Star").
I will let the experts comment further.