Inspired by what I DIDN'T like in a favorite book

Recently I read the whole Hunger Games trilogy on Kindle (because it was cheap, and the story sounded interesting). I liked it. But what inspired my writing side was not the good parts of the book, but something I didn't care for.

Panem, the world in which the Hunger Games takes place is a future United States, but it is also a parallel to the ancient Roman empire. With one big difference between it and its sources. In Panem, there are no Christians. Or Jews, Muslims, Hindus, Wiccans or Pagans, for that matter. I felt the lack was illogical.

This inspired ideas about an unfallen Roman empire in which Christians existed but never became dominant, and were subjected to persecutions including the possibility of death in the arena. My story was centered around the idea that a group of young people--- children of Christian parents--- were ordered to fight to the death in the arena.

The unfallen Rome of the story I named 'Kirinia' and decided it was another world, that ancient Romans had gotten to by means of a Portal (transportation gateway between worlds/locations). I got the name 'Kirinia' from the name of an obscure Roman god, Quirinius. The Romans were actually sometimes called 'Quirites' from that name. I changed the spelling to eliminate the 'q'.

The name 'Kirinia' reminded me of the 'kirin', an Asian mythological beast which played a part in a world I created called 'Erileth'. This world had 5 kingdoms which corresponded to the Asian 5 Elements (wood, fire, earth, water, metal). Each element had colors, directions, creatures, livestock, grains and the like which corresponded to it, and so I decided to use these as a basis for creating the kingdoms.

The Earth kingdom in the Center of the world had the kirin as its beast. When I consulted my Erileth binder, I saw that I had decided to base the Earth kingdom's culture on ancient Rome. That decided me on the project of merging the new story's world with that of my old Erileth/5 Elements world.

Anyway, I got to wondering: what is it about our favorite books that inspires us with more/better story ideas of our own--- the parts we do like, or the parts we don't? In YOUR experience, what inspires YOU more???