3e did 1 1/2 x Str mod, right? So a Str of 20 would be +5 and 1/2 more again would be +7?Depends on your edition. 5e eschews modifiers and advances die type so going to 2d6 may be appropriate.
2e generally keeps dice the same but adds +2 damage. 3e gives a stupidly good multiplier to strength bonus damage (particularly when strength is really high).
1e did it really weird. I won’t even go into it here.
Although it changes the question parameters, I would say 1d8 for 1 handed and 2d4 for 2 handed. Mathematically, the halfway point of a d8 is halfway between the 4 and 5 (the imaginary 4.5). On a d4, it's the imaginary 2.5. Double that 2.5 and you're at 5. Which is .5 higher than the imaginary 4.5 of the d8. It's a slight boost but not a gamebreaker. Practically, the lowest possible then becomes the 1 when used 1 handed and 2 when used 2 handed.
Longsword(historically) is a two handed weapon, that you could use one handed in some moves(i.e. lunge), but you would want to use it two handed almost all the time.The 5e longsword is a bastard sword (as it was RL historically).