Can I get a refresher here? Are you talking about Jaquen? Was it known he was there until Arya was on her way north?
Well, we know Jaqen was in the Black Cells at the same time as Ned since Yoren said that is where he got Jaqen, Rorge, and Biter from.
Here is where the theory starts:
Faceless men are the world's premiere assassins. From what we have seen of Arya's training and Jaqen's abilities it seems highly unlikely that he was just caught and tossed in the dungeon. It seems far more likely that he was put there for a reason.
Going off of that premise - who would be Jaqen's potential target? The only person in the dungeons would be Ned. Also, the plan for Ned once he confessed to being a traitor was to go to The Wall with Yoren - so Jaqen is perfectly placed there as well. So - who wants Ned dead and who has had contact with Faceless men? Littlefinger.
Littlefinger hates the Starks - has for a long time. We also know that Littlefinger has approached/had dealings with the Faceless men (he talks about their prices when the Small Council is discussing killing Dany). We also know that Littlefinger's family was initially from Braavos (where the Faceless men reside). On top of that we know that Littlefinger wants chaos. Assuming that Littlefinger wants Ned dead it seems likely that he is the one that influenced Joffrey to execute him (and might have told Slynt since we know Slynt didn't hesitate when Joff called for Ned's head). However, if Cersei, or Varys, or anyone prevented the execution from happening - he has the backup plan of Jaqen going to The Wall. Jaqen is also very skilled about making deaths look like accidents. I wouldn't be surprised if Ned's death was to look like an accident or to look like a Lannister killed him.
This would also explain Jaqen knowing who Arya was when he came across her in the Harrenhal godswood.
Does this theory then continue with Jaqen / Pate in Oldtown?
Not really - or at least not unless there is some evidence that Littlefinger wants something/someone at the Citadel of Oldtown - which I don't think we have directly read or seen. We know Littlefinger is preparing for Dany but making a leap that he wants something from the Citadel seems a bit much without any proof. As it stands here is what I think happened.
1) Littlefinger hires a Faceless man (as detailed above) who turns out to be Jaqen
2) Jaqen, while waiting to escape, has his life, Rorge, and Biter's saved by Arya.
3) Jaqen's contract with Littlefinger is over, he now needs to get in contact with Braavos
4) Jaqen, Rorge, and Biter join the Brave Companions - likely to save their skins, but Jaqen's iron coin might have come into play as well... who knows how this happened.
5) At Harrenhal, Jaqen now has the means to contact Braavos (raven or messenger) - but also meets Arya - to whom he owes a debt.
6) Jaqen, while waiting for a response kills Arya's first two targets (she searched him out both times), once receiving his orders, Jaqen tracks down Arya to ask for her 3rd name so he can fulfill his obligation to her and move on to his next mission.
7) Arya blackmails Jaqen with his own name to help her. Once the northmen are freed then Jaqen gives Arya his coin, changes his face, and moves on to his next assignment.
Now, I would have to check the timing in regards to Littlefinger see if it would still work out if you replaced "contact with Braavos" with "contact with Littlefinger". If Littlefinger was away from King's Landing (going to the Tyrell's) I think we would have to assume that Jaqen is no longer working for Littlefinger at that time.