Telekinesis, Eyebite, and Bigby’s Hand are some of the few spells which persistently affect one creature. As such, they appear to fit the requirements of Twinned spell, which are…
Twinned Spell: When you Cast a Spell that Targets only one creature and doesn’t have a range of self, you can spend a number of sorcery points equal to the spell’s level to target a second creature in range with the same spell (1 sorcery point if the spell is a cantrip).
To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level. For example, Magic Missile and Scorching Ray aren’t eligible, but Ray of Frost is.
Bigby’s Hand, Eyebite, and Telekinesis are all incapable of targeting more than one creature at a time at any level. This leads to…Difficulties…interpreting then. Can you even twin them? If you can twin them, how do you target them on future turns? Can you cause different effects to different targets on each turn, per Eyebite or Bigby’s?
I included Bigby’s because of multiclassing.