First off a HUGE thank you for writing! Overall, I'd really like if you have fun with this story and write something you'd like to read yourself. Stories that make me happy tend to concentrate on the characters being in character and acting and speaking like they would on the show and lean towards the gen side of things. But overall, I love all three of these fandoms and would be thrilled to read any story written for them.
New Amsterdam (tv) John Amsterdam, Omar York
Father and son interaction between these two characters, please. Either subtly like we saw in the pilot when John gets Omar out of trouble with his bookie by making a desk or something more overt. Or perhaps, something about the problems of an immortal having a child, and the child gets older and older while the parent doesn't age? Gen please.
New Amsterdam only lasted eight episodes, but it should have been more. Luckily they are all up at Hulu so they didn't disappear completely. I loved how at first we thought Omar was just a bartender and a friend of John's, but then we learn differently. It's also great that in the flashback with Omar's mother, Lily, we learn that John's secretary at the time was (presumably) also his daughter. It's so rare for immortals in fiction to be able to have children (as seen in most vampire myths, the Highlander series, etc.) that dealing with this issue is fascinating to me. What was it like to have a father that was younger than you? To see your child grow up while you don't? Obviously John and Omar figured out their relationship, and it's such a wonderful one. I just love to see them interact.
Big Valley (tv) Heath Barkley, Victoria Barkley
Canon never told us exactly why or when Heath started calling Victoria "Mother." A story about that moment or what led up to it would be wonderful, but any story about the relationship between these two would be absolutely fantastic. Gen please.
Luckily S1 of Big Valley is also available at Hulu because it doesn't air too often anymore. I was always amazed how gracious Victoria was with Heath, she truly came to consider him one of her sons. But how? What made her open her heart up to him in that way? Why did he go from calling her Ma'am to Mother? How did this new family and new mother affect his feelings towards Leah and Rachel and Hannah?
Haven (tv) Dwight Hendrickson, Nathan Wournos, Garland Wournos
How did Dwight start to work for the Chief? What were some of the situations he 'took care of' during the Troubles? How was working for Garland different than working with Nathan? Did he like his job? Just how many bullet proof vests does he own?
Or, if you don't feel comfortable writing about Dwight, then something regarding Nathan and Garland's relationship, please. They had such a great dynamic, and there was such tension, yet love and respect between the two. Did Garland ever wish he was just "Dad" rather than being "Chief" so often? How did his role in holding together the town affect his relationship with Nathan? Did he ever think of Nathan as Max's son and not his own? Gen please.
I offered the two different ideas here because, while I love Dwight, I realize he may not have appeared often enough for someone to feel comfortable writing him and I'm also fascinated by the relationship between Garland and Nathan. Haven is a great show and I'd really love any story about any of these three characters.