The first people that came to mind when I saw this thread were Littlefinger and Varys, then I put some thought into it. Benjen(If he's actually alive), The High Sparrow, Howland Reed, Pyat Pree, Thoros, and The Blackfish all had potential as well. In the end the one character I would ask for would be Quaithe. Probably the most mysterious character in the series.
All the other characters I've mentioned,with the exception of Howland Reed, we've interacted with.
We know about the Baelish families rise to nobility, and of Littlefinger's childhood, his obsession with Cat,he's two faced, manipulative etc. We may not know his endgame, but we certainly know a lot about him.
We know Varys' alleged history, personality, some of his abilities(Master of Disguise) and get to see some of the plots he is involved with.
We know Thoros' story, how he lost his faith, then preformed a miracle. We've seen some of the fallout from that.
The sparrow is a fanatic with an army, we don't really need more than that...mind you he could be Howland Reed, But that's another thread entirely. I also just realized the lack of King's Landing POVs currently...Is it really just Cersei now?
Brynden Blackfish is someone we've seen many times. We know his history, his personality, that he's kind of a badass. We may not know his current whereabouts or actions, but we know he will return at some point.
GRRM has outright said we would meet Howland Reed at some point, I'm willing to be patient. Benjen could be dead, can't risk wasting a request. If Pyat Pree is one of Victarion's captive warlocks, then we already have a POV close enough to give us some insight.
Quaithe. We've interacted with her, yet know literally nothing about her. We know Name, Gender, Description. She was an emissary of Qarth, but "of the Shadow" implies Asshai. Is that where she is originally from? Or did she travel to Asshai from somewhere else first. Is she a shadowbinder, or does she just have an active glass candle, or is it a combination of the two? Why does she try to,albeit cryptically, help Dany? Is she still in Qarth, or did she just show up there to meet Dany? She seems to have access to information, how is she obtaining it? Glass Candle? Divination? Spy network? something similar to Bran's Wierwood net? Who is she really, and what is she capable of? What are her motivations? What is her part in the song? Is she working independently, or is she part of a group? So many questions, so few answers. We get some cryptic prophecy, and that's it.