Twinned Spell says:

When you cast a spell that targets only one creature and doesn't have a range of self [...] To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level.

Does hex meet this definition?

While hex initially targets only one creature, later on it explicitly allows you to choose a new target which could mean that it would be targeting more than one creature. And Twinned Spell does not seem to limit the restriction on targeting to any given time frame.

You place a curse on a creature you can see within range. [...] If the target drops to 0 hit points before this spell ends, you can use a bonus action on a subsequent turn of yours to curse a new creature.

Does this ability to choose a new target disqualify hex from being used with Twinned Spell?

You can't twin hex

Twinned spell states (as of the first official errata):

To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level.

Hex is able to target more than one creature if the first one dies so it is ineligible. This targeting is referred to as "curse"ing but the effect is the same.

A typical spell requires you to pick one or more targets to be affected by the spell's magic.

The creature you are selecting to be hexed is the target of hex, so the new creature is another target of hex. The word target never appears in the text replaced by the action of cursing (emphasis mine):

You place a curse on a creature that you can see within range. [...]

If the target drops to 0 hit points before this spell ends, you can use a bonus action on a subsequent turn of yours to curse a new creature.

As such, the hex spell counts as targeting more than one creature. While at any given time only one creature is affected, multiple creatures may be targeted during the duration, so it can't be twinned.

This is reinforced by Jeremy Crawford, lead designer, where he talks about the telekinesis spell on twitter:

Telekinesis can affect multiple creatures over the course of its duration. It's not eligible for Twinned Spell.

This is a comparable ruling because telekinesis, like hex, only targets one creature at a time, but can affect multiple creatures with one casting.

You can affect the same target round after round, or choose a new one at any time. [From telekinesis]

It depends on how you define "targets only one creature"

When you cast a spell that targets only one creature and doesn't have a range of self [...] To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level.

To answer this, we need to drill down and answer thoroughly what a "target" is in 5th edition D&D.

The Crawford School of Linguistics proudly declares NO!

Jeremy Crawford specifies that spells like Dragon's Breath, for example, "can affect more than one creature", and is therefore ineligible for being Twinned. This implies that to Crawford, a spell "targets" anything it can affect. So in this reading, Hex, being able to "affect" more than one creature, is ineligible to be twinned.

The Formal Reading of Spellcasting Targets begets Confusion and Ambiguity

In the Player's Handbook, Chapter 10 "Spellcasting", subsection "Targets", the concept of a spell's targets is described like this:

A typical spell requires you to pick one or more targets to be affected by the spell's magic. A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area of effect (described below).

So here's how I read this:

I go to cast Dragon's Breath. Dragon's Breath targets one creature: the creature I'm placing the "buff" upon. The "buff" on that creature permits it to deal an AOE damage effect, but at no point during the casting of the spell did I, the person casting the spell, target more than one person.

Therefore, the spell targeted only one person, and is eligible for being Twinned.

But this wouldn't make Hex eligible, because you choose the secondary/tertiary targets

Under this reading, the spell only becomes eligible if the spell itself does not target an additional creature. Hex still grants the ability to target an additional creature (and specifically grants this capability to the caster) so it remains ineligible

The School of Simultaneity Issues an Injunction and says YES! (but isn't very convincing)

Hex might be able to affect more than one creature, but it only affects one creature at any given time. One possible reading of Twinned Spell is to argue that because Hex cannot affect multiple creatures at once, it is therefore eligible to be Twinned.

This is a bit of a stretch though, because it requires a very specific interpretation of "targets only one creature" to infer "at once" at the end.

The School of Postmodernism asks: what even is a "target", really?

Does Hex "target" a single creature, and all other effects are tertiary and not "targets", or does Hex "target" a single creature, and then "target" a second creature, and then "target" a third creature? Does "target" require the word "target" to actually appear in the spell description when specifying secondary effects, or does the phrase "you can use a bonus action on a subsequent turn of yours to curse a new creature" imply the spell is not "targeting" another creature? Does the word "target" seem weird to anyone else? Are words even real? Does anything actually exist?

The School of Pragmatism says: Probably not

Relaxing the definition of "target" to mean "anything affected by a spell" is, to me, a bad precedent, which is why I do not agree with Crawford's ruling on Dragon's Breath, or on a large number of other spells potentially valid/invalid for being Twinned. I suspect this may have been made on Balancing implications, not on clarity/conciseness implications, but to me, when I target a spell, I choose whom it applies to.

At the time I cast Hex, I'm targeting one creature. At some later point, I can transfer the spell by targeting someone else. So I think it's fair to say that ultimately, Hex cannot be a valid spell to be Twinned. But other spells like Haste, Dragon's Breath, etc. where I'm only "choosing" a single creature to be affected, should be valid targets, even if the outcome of their effects is to affect more than one creature.