"He hopes that he likes it."
Translation:Ελπίζει να του αρέσει.
I am not a moderator but I have the same question and since the question was asked 9 months ago, I'll give it a try and then maybe a moderator will chime in. According to Peter Mackridge's 16 page section on να in his The Modern Greek Language (pp. 282 ff), "The use of να is so frequent and so varied that a systematic and comprehensive survey of its uses would be beyond the range of this book." That leads me to think that DL here is introducing us inductively to one of the ways να works. It may be helpful to know that να is a marker of the subjunctive, and since we haven't gotten to that mood yet, clearly we are in inductive learning mode here. Mackridge continues: "In many of its uses, να + verb is the equivalent of an infinitive in modern Western European languages. The lack of an infinitive, which is periodically lamented by Greek writers, makes it impossible for a verb to be used without at least an implied subject... On the other hand, the flexibility and versatility which να puts at the speaker's disposal provide a degree of compensation for the lack of the infinitive." Finally, Mackridge provides examples of usages of να and one of them is in Wishes and Curses referring to future time (284) and another is "volative" (286). The latter is a grammatical term that refers to desires, wishes, and fears (Latin volo, "I wish"). Ancient Gk used an infinitive after verbs of hoping, expecting, promising, threatening, swearing when they refer to a future event (Smyth, 418), but modern Gk [MG] doesn't have an infinitive so the other way of handling the matter is through να. According to Horrocks (Greek: A History of its Language and Speakers, 297) "In the later middle ages να + subjunctive largely replaced the infinitive." Horrocks (228) also notes that beginning in the Byzantine period να + subjunctive to mean "wish" was used along with να + subjunctive to mean "will" (the latter indicates that να is a short form of AG ινα) but the latter gave way to θα as the marker of futurity (AG θέλω > θε να > θα, p. 229). Anyway, MG uses να here as a subjunctive marker governing a subordinate verb (see Mackridge, 279). Since the first verb here concerns hoping, it would seem to require the subjunctive mood, whereas ότι takes the indicative and is typically used after verbs of showing, perceiving, knowing, believing, etc (Mackridge, 269). An example from Mackridge's grammar (270): είμαι βέβαιος ότι..., "I'm sure that..." is an indicative statement (a fact). Blah, blah, blah.
Looking at the incubator, it seems that "I hope that she likes it" is translated as Ελπίζω να της αρέσει, without αυτό. Maybe you encountered a translation now removed, it happens sometimes. With the usage of αυτό, the sentence means "I hope that she likes this", not it.