In what way would you expand the story?
So when I heard Breath of the wild had memories I thought 'great!' these would be like Halo 4s terminals and would show the past in an interesting way. If you haven't played halo 4 just take my word for it that the terminals offered exciting glimpses into the history of the Halo lore. I would see a similar system implemented into OOT HD. IIwas thinking maybe gossip stones would be the way to access these cutscenes.
Anyway each area you visit might have a stone for instance and the cutscene the stone offered would relate to the history of that area or the people's connected to it. The history of the Sheikah could be shown in Kakariko and explain why there are so few left. One in Desert Colossus might show a younger Twinrova and go into how they have been operating behind the scenes for hundreds of years and how they came to be surrogates for Ganondorf perhaps.
As well as the stone cutscenes, characters who had just a few lines before (so most of them) could have proper sidequests made to incorporate them, Witcher 3 style. There you really get some depth and development and get a proper feel for what these characters place in the world is.