Not sure it would work. It kind of sounds like some would think the 'son' idea what auto-work but we don't know how that would work or if the character would be as interesting. Maybe the best choice would be sticking to the AI since it would mean he could sort of exist in his own way without messing with lore much.\
The son wouldn't auto-work, but I think it'd fit the best if people want a 'johnson'.
I don't see how an AI would be any better. Smart AI's are not perfect copies of their originators, they sometimes have some quirks and traits but they're their own person. Cortana isn't Halsey, she shares some similarities but she's not much different than a daughter.
You also have the lore issue of how they made an AI of johnson. It's unlikely his brain survive falling down a giant shaft, into a charging halo beam, on an exploding halo, then sitting around for a few months/years in a preservable state for someone to come back and use it to make a smart AI.
The AI would just be an excuse to have him in some form...it would help give an excuse for having his voice become canon in multiplayer via voice.
Of course Halsey isn't Cortana but that's because of the way she was designed and if Halsey wanted to then she could have easily created a Halsey AI that believed she was the same person. Look at Star Trek for example...we have AI like the Doctor that is different to Lewis Zimmerman (creator)
with a very different personality while they have Holodecks that have AI that are mostly a closer resemblance to their original counterpart. Plus if I recall I believe the multiplayer was supposed to be canon being a sort of Holodeck. In some cases the AI could have a more closer personality with minor changes like Vic Fontaine hologram
from ds9 where the personality was more like his original counterpart, yet still made his own decisions. I'm sure the technology could be improved and they could fabricate the personality into an AI.
As for Johnson's son I'm not sure it would work...really depends on the personality of the character. It seemed to work with Miranda Keyes but don't think everyone's son/daughter would automatically join the military.