I am too lacking in Korean music knowledge to come up with the perfect song, but this is such an interesting question it got me very curious.
There are so many great songs of that time, especially ballads, that I don't think the producers need to resort to remaking Last Dance or even creating a new song. Personally, I think it has to be a real song from that period so that the audience can relate to it, and bring out the longing and nostalgia that comes from music of the times. The Reply series and even Hospital Playlist wouldn't have the same feel if all its music was new/written for the show.
I would love for the producers to go for a similar rock-blues feel, but as far as I know this style of music wasn't as mainstream during that period in Korean (which maybe could work in its favor). I came across a rock-pop-ballad by YADA "Already Sad Love" which I thought gave the same heartbreaking feels as Last Dance, especially in the chorus, and now I can't stop imagining it as the song ^^. (Though its lyrical content might be too on the nose.)
- YADA (야다) - Already Sad Love (이미 슬픈사랑) - https://m.youtube.com/watch?v=Vz8GBHf1mq8
If not the rock-ballad route, I think a song from an established ballad artist like Lee Sun Hee could also work. "Always Miss You" has a lovely bluesy feel.
- Lee Sun Hee (이선희) - Always Miss You (나 항상 그대를) - https://m.youtube.com/watch?v=bqvPL5zuJmA
Some other contenders:
- PANIC (패닉) - Snail (달팽이) also came to mind, but it was used in the Prison Playbook OST so unlikely to be used... https://m.youtube.com/watch?v=wNPY1lSo8Mc
- Shin Seung Hun (신승훈) - Reflection of You in Your Smile (미소속에 비친 그대) - https://m.youtube.com/watch?v=k4X0z_LvDDs
Note: I picked out some songs based on their popularity and feels, not necessarily fitting the exact year.