I was wondering what a lance made of mithral from the 5th edition would do. Mithral weapons are lighter than other kinds of weapons, but I read on what mithral weapons do, but it never explains what mithral weapons do with the lances special property. I don't know if I'm answering my own question, but would the lance have the versatile property since it's technically two-handed? If so, what dice would you roll for using the weapon in one hand?
There are no RAW specific effects
Unfortunately, there are no rules about either coating weapons in mithral or entirely mithral weapons.
The only mention of mithral effects are from Mithral Armor.
If you/your DM are interested in weapons of mithral, you'll be in homebrew/houserule territory (which is perfectly fine...but it'll be up to you.)
Versatile vs two-handed
This is sort of a second question hidden in there, which probably should be it's own..., but if a weapon property isn't listed, it doesn't have it. Having the versatile property isn't the same as the two-handed property. They aren't interchangeable as they have different functional mechanics.
The lance, by default
The description of Lance says that it deals 1d12 piercing damage. It doesn't have the versatile property but the special property instead. Regarding how many hands are required for wielding a lance, the weapon description continues with:
Also, a lance requires two hands to wield when you aren't mounted.
So when you are mounted, you could wield a lance with one hand, but—whether you wield it in one or two hands—the damage die stays the same for this weapon.
The lance, coated with mithral
As far as I know the Mithral property only applies to armor, such as Mithral Armor, but not to weapons. It sounds like an interesting idea though, so it's something you could suggest to your DM. But keep in mind that this would be homebrew terrain, and not all DMs want that at their table.
Neither RAW nor Necessarily "Realism" Make it Lighter, but we are in Homebrew Magic Item Territory
RAW does not have mithral weapons, only mithral armor. Even with the armor, however, there is no actual mechanical reduction in weight, despite the obvious lore reasons for having such. Crawford has tweeted to this effect:
Mithral armor doesn't weigh less than normal. Its description would say if it did.
Hard to disagree with that principle of rule interpretation, even if here it is directly inconsistent with the lore of being lighter. They could have easily said "any mithral armor will weighs half as much as it normally would", or some such, if they intended that to be a rule, and there are no secret rules.
And, once again as others have said, once we are making mithral weapons at all we are already beyond RAW.
"Realistically" It Probably Shouldn't Do Much for a Lance Anyway
Within the degree we can expect "realism" for a magic metal in an elf game with many mechanical conceits, remember that lances are made of wood (often with a metal tip), not primarily of metal. The relative weight compared to wood is not remotely alluded to in the lore, so we are in even more uncharted territory than with a metal weapon.
Beyond material concerns, any similarly long object is fairly unwieldy in one hand. The exception for a lance while mounted is based on the real world fact that with the horse providing the striking force and much of the direction of the lance it can be couched under an arm for stability. In other words, if you are planning to give it a normal one-handed melee option that probably doesn't make a lot of sense even if it is lighter.
But, once again, mithral exists in a world of magic and wonder, and having it bestow all manner of magic properties on a weapon made of it makes sense enough. I'd simply argue that, for a lance, this really has no strong logical connection to the lore of it being lighter than normal metal. Make it magical however you like.
Homebrew-Wise, There does Seem to be a Consistent Pattern to Guide You
It seems you are thinking the mithral should give it the versatile property, effectively extending its special property of mounted one-handed situations. There is no game rule to support this so we are in the realm of homebrew, which is fine. Some considerations:
As I'm guessing most people have noticed, the general rule of versatile weapons is that they use one die lower when used one-handed than one used two-handed. If you are interested in homebrewing it then this would probably be the most natural and RAW consistent approach to take when giving an otherwise (usually) two-handed weapon the versatile property.
Obviously a one-handed reach weapon doing 1d10 is probably a pretty strong item requiring a healthy dose of rarity. Normally the only one-handed reach weapon other than the lance while mounted is the whip, which does 1d4 and has no two-handed option (it does also have finesse though).
In any case, if you are going to homebrew the versatile property into a mithral lance, these seem like relevant considerations.