Hum... I think maybe OD&D had some jousting skills? Or chainmail?
While playing GURPs, I usually had only two checks: a "lance" check to see if you hit and a "ride" check to see if you stand.
This wouldn't be too hard to do in 5e, IMO. Let's see:
1 - Some kind of ride" check, DC 20. If successful, you can get advantage on step 2 or 3 (your choice).
2 - Attack roll with lance.
3 - Make a concentration check to see if you fall from your horse.
Of course, there are LOTs of caveats here.
This isn't great for fighters, since they do not get to use their multiple attacks. Rogues could apply sneak attack with advantage, which would make them the best jousters if they have enough Str.
Probably allowing super-heavy armor or limiting dexterity bonus to AC would work well.