For me, the one necessary component is that I have to really like the song. Everything else can be great, but if I don’t like the song, it isn’t going to stick with me. When it comes to visuals, I am not too picky, but I do like it when shows get experimental with things. I think that is one of the reasons why things like Kaiju and Salad Bowl OPs ended up on my list, due to how different they looked.
I am definitely influenced by how I feel about a show though. It usually is more impactful when the OP/ED is used as part of the storytelling. I mentioned elsewhere in this thread that Jellyfish is one of those shows where, if I was watching it, I can imagine the EDs that they are making would be more impactful. I also think that Konosuba benefits in this regard since I watch and enjoy the show.