Why is J the only person who remembers K?

He remembers agent k because he (agent J) was there--as a child. The pawn shop guy that helped him time travel forshadows this when agent J asks why he remembers.

It is because right as the time fracture happened he had just been in contact with K and K was heavily on his mind. If he had been doing just about anything else besides having just gotten off of the phone with K then he probably wouldn't have remembered him.

I have been wondering that same thing, I can't quite figure it out. I have two theories though. Either it had something to do with the fact that he time-traveled while in the past, when he jumped down from the rocket launch thingy with Boris. Maybe that somehow made him resistant to time fracture, because he wasn't really present while the change happened. Or maybe it has to do with agent K neurolizing young James. It somehow made him forget about the changes in the past.

I realize it's not very helpful, but that's the best I have been able to come up with.

Jeffrey says “you were there” and seems excited to know the details. There are three possibilities:

1. He was present (at the launch) when Boris killed K. Somehow being present at the point of the time break makes him immune to the rewrite. Jeffrey is excited to hear about the launch from someone who was there. It’s a dumb explanation but just dumb enough to fly in a Hollywood movie.

2. He was the one who last had contact with K just before K was erased. Somehow being in close contact with someone who is erased makes you remember them after the rewrite. Jeffrey is excited to hear what it was like to experience history being rewritten around you. Again, dumb but Hollywood.

3. J was also present at some unspecified major phenomenal cosmic event and it renders him immune to time rewrites. Jeffrey, who somehow knows about the event is excited to hear about it. Possibly dumb, possibly smart, depends on the exact explanation. Possible setup for MiB4 which depicts more time travel, and seeding sequels is extremely Hollywood. Another time travel storyline set between 1969 and 1997 might explain how O already knows about chocolate milk being a thing, since J never mentions it to her or K in 1969. It would also give the studio an excuse to use Brolin again instead of the aging Jones, and Hollywood loves going young.

The real answer is “nobody knows”, unless the filmmakers tell us. Honestly, the movie is full of holes (Brolin is supposed to be in his twenties?!), so there might not even BE an answer.