"Eleni has the book."
Translation:Η Ελένη έχει το βιβλίο.
it would even be incorrect to form this sentence without using Η .
"Ελένη έχει το βιβλίο " is gramatically incorrect
The rule is that proper nouns always need an article as subjects or objects of a sentence because it sounds weird otherwise. Not using it is equivalent to saying in English "car has broken down". The only case I can think of that you can use a proper name without an article is when it is a predicate in the sentence, as in "Το όνομά μου είναι Ελένη", meaning "My name is Eleni".
This question is really strict with the accents. "Η Ελενη εχει το βιβλιο" is not even accepted.