well with the direction this thread is going it would seem everyone in here so far agrees w/ the showrunners that Jon is the son of Rhaegar and Lyanna. I don't actually subscribe to this theory as it's been widely accepted for to long and despite what most people think about the showrunners I do think they are trying to make the show as different from what the future books will be as possible and the R+L=J theory was the most convenient one. Also they saw the disaster that Dorne was and probably didn't want to introduce anymore characters from the region (like ALL the Daynes except Ser Arthur). If I'm correct it would mean a huge divergence from the books and show that anyone who cares to read the books from here on out will know the actual parentage GRRM intended (when he gets around to writing them, if he does choose to throw this curveball) that with the now huge audience he has he will still be able to surprise people rather than the show spoiling everything in the last few seasons because they ran out of material. We all know that GRRM's test for the showrunners was who they thought Jon's parents where, so if they decided to change it, right now it's only them and George who actually know
to answer the question posed by the thread, If I was Ned I would have probably told Cat that Jon was the legitimate heir to the North by Brandon and Ashara (who probably got married in secret while he was imprisoned at the red keep)
lets look at the books and early show and how it treats DNA, you would think Targ features would be recessive to ALL other family features but that is not how it works in this world, according to the book of linages in the first season All children born in the Baratehon line had the same features which is how super detective Ned and co (Jon Arryn) found out that Robert didn't have any children w/ Cersei. We know from lore that Lannisters have married into the Baratheon line before and their features remained unchanged, same w/ Baratheons marrying into the Lannisters, the fathers genes always seem to be more dominant over the mothers. The only time this gets somewhat muddled is when the Targs and Martells intermarry and even then only like 2 members of the Targ dynasty are ever described as having anything but Targ features and those where Baelor Breakspear and Rhaenys (Rhaegars daughter) so going by the early show and book logic if Lyanna and Rhaegar had a child it would look like Rhaegar meaning it was most likely Danny or even more far fetched Aegon. On the other hand this also supports if Brandon and Ashara had a child it would look like Brandon making it easier for Ned to hide (important note: Daynes have similar features to Targs). So according to book and early show logic R+L=D and B+A=J
to support this, why would Ned stay in Kings Landing after quiting the small council about Robert having an assassin murder a Targ girl that had just been married off to a horse lord when he started having fever dreams about his promise to Lyanna and the Tower of Joy, dreams that ultimately made him decide to stay in Kings Landing and lead to his death as a result when he knew Jon was safe at the wall and had renounced all his claims to anything when he joined the Nights Watch? The only person who was a known Targ and was even in the realm of possibilities that Ned might be breaking his promise to Lyanna was Danny at the time.