Thanks for confirming that, when Duolingo makes mistakes in English I am quick to report it, when it gives what I think is the wrong response for a Portuguese answer I doubt myself first. You're right though "What could have brought it here?" does sound like a line from a science fiction film :-)
Here's how I imagine the sentences being used: you're at a party and suddenly a stranger comes in, so you wonder "What could have brought him here?" ("O que o terá trazido aqui?"). The guy leaves and you're still thinking "I wonder what could have had brought him to the party?" ("Eu me pergunto o que o teria trazido à festa").
Since the tenses overlap, you can also use them in a more literal way (future tense and simple past, for example), but in my opinion, that's the most common usage, even if it's not grammatically correct (I don't know if it is or isn't actually).
To teach the Future Perfect, it would be useful to have two future actions in the sentence, one taking place before the other.
The movie starts at 9 pm, but we can't get there before 9:15.
• The movie will have already started before we arrive.
• O filme terá começado antes de (nos) chegarmos. Correct?
Erudis, I agree that the English sentence you identify seems very familiar in usage. However, as emeyr points out, in your party context above, it would be more like, A guy (stranger or not) says, "I'm going to leave and come back either by motorcycle or horse. Why don't you guys bet on which one it will be? You turn to some other people there and say, "What will have brought him here?" It's an approaching realistic future reality in that context, not more hypothetical in that context like the phrase "could or would."
At first sight it wasn't obvious to me why this sentence is written "o terá trazido" and not "terá o trazido" (putting the pronoun between verbs like this seems to be the Brazilian default even if it is not endorsed by strict grammarians).
It turns out there is a rule that says certain words ("palavras atrativas"), such as "que" and "não" attract the pronoun. This is mentioned briefly in this note dealing with pronoun placement: http://www.infoescola.com/portugues/colocacao-pronominal-proclise-mesoclise-enclise/