Questions & Answers about Hún lofar honum að koma heim.
Why is it honum and not hann?
The verb lofa (to promise) requires its object to be in the dative case. Since the basic word hann (he) changes to honum in the dative case, we use honum to mean "him" here.
Why is there an að before koma?
In Icelandic, að is used just like the word "to" in English when connecting a conjugated verb to an infinitive (dictionary form) verb. Because she promises "to come", you use the marker að followed by koma.
Why is it heim instead of heima?
Icelandic uses different words for "home" depending on whether there is movement. You use heim when someone is going or coming home (a change of location). You use heima when someone is already at home (no movement). Since she is coming, we use heim.