At the time that this is being written, we don’t know if there will be a Halo 2 remake. If there is one coming, there will obviously be additions such as terminals. Here are some guesses as to what they might be, while keeping in mind how they affect the story itself. Feel free to add your speculation as well. 
Mission 1: Cairo Station
There are two mysteries that came from the Original Halo Trilogy that were never explained in depth. 1) The Covenant’s discovery of Earth, and 2) How the Flood discovered the location of the (lesser) Ark in 2552. In terms of Earth’s discovery, it may have been due to Luminaries on board covenant starships, and in First Strike, an entire fleet was supposed to head there. Obviously, only fifteen ships arrived, not expecting the planet to be the Humans Homeworld. Hence the first Terminal, Regrets ambition and how it affects Truths plans for “The Great Journey”. (Truth and Regret are the only ones in the covenant leadership who know the truth about humanity, keep this in mind)
Mission 2: Outskirts
Guessing this one was a no brainer.
In Halo 3: ODST’s Sadie’s Story, Dr. Endesha points out that ONI believed the Covenant were looking for something under the city. It was also said that during a seismic survey, an archaeological anomaly was discovered underneath New Mombasa. This is what I believe what the second terminal in the game will be about. It won’t be an ONI committee listening to Endeaha speak, but rather the Librarian giving her thoughts on the construction of the Ark portal itself. (With a time lapse that shows the portal being buried by the sands of time, and the growth of New Mombasa right over it)
Mission 3: Metropolis
This terminal should tie in with the first, here’s how.
In “outskirts”, it’s mentioned that the covenant started pulling back to the city center. As we all know, Regret was preparing to retreat the city via slipstream space, and from the ending of ODST, more ships arrived to finish what was started. The third terminal explains how Truth’s plans to shake up the covenant leadership starts, with him sending a fleet of brute controlled ships to exterminate the sangheli on earth.
Mission 4: The Arbiter
As we all know, the Elite known as Thel Vadamee was turned into the Arbiter. So, the forth terminal should explain how truth plans to use the Arbiter as a puppet, in order to set bigger things in motion.
Mission 5: Oracle
The 11th terminal in Halo Anniversary explained how Guilty Spark and The Heretic joined forces. The fifth terminal of this game should continue on where that terminal left off, specifically why Flood specimens were on the Threshold Gas Mining Facility.
Mission 6: Delta Halo
The terminal for this level should be about Installation 05 itself, after all, the story just revealed its existence in this level. The terminal should explain what happened to the monitor Penitent Tangent, and why a gravemind exists on the ring prior to the Flood’s release in the game.
Mission 7: Regret
As we all know, John killed the High Prophet of Regret in this level, even after " the largest covenant fleet…anyone’s ever seen" arrived at the ring. Later in the game, it’s revealed Regret could’ve been saved had Truth not withdrawn phantoms that were headed there. Terminal 7 reveals how John’s actions lead one step closer to the betrayal of the Elites, and how it seals the Arbiter’s fate.
Mission 8: Sacred Icon
Plain and simple…
The 8th terminal shows Tartarus and Truth having a conversation. Tartarus is ordered to kill the Arbiter as soon as the Icon is secured, and to start the (attempted) extermination of the elites.
Mission 9: Quarantine Zone
(A break from the diabolical Truth)
Commander Miranda Keyes, a product of what happened between Dr. Halsey and (at the time) Lt. Keyes during the time that they surveyed Spartan 2 prospects. At this point in the game, UNSC Amber Clad is infested with flood and most of her crew are dead or infected. The ninth terminal focuses on Miranda and her thoughts about how she could ultimately face the same fate her father did, death or infection by the Flood.
Mission 10: Gravemind
The second mystery begins to unfold.
The cutscene that starts this level off reveals something interesting about Regret. He is now apart of the Gravemind, the Flood’s central intelligence. The terminal in this level explains how regret found the location of The Ark, and shows the Gravemind’s thoughts about what happens if the Chief and Arbiter succeed in stopping the ring from firing.
Mission 11: Uprising
This level is the culmination of Truth’s plans. Terminal #11 goes into the thoughts of R’tas Vadamee (who will become the “shipmaster” in Halo 3) and shows vivid images of Elites being Massacered in what appears to be a losing effort.
“They have shed the blood of our brothers…and for that they must die!”
Mission 12: High Charity
The terminal shows the Gravemind teleporting to High Charity, as the Flood take advantage of the Covenant Civil War. This scene also sets up the plot for Flood’s arrival at the ark in Halo 3’s storyline.
Mission 13: The Great Journey
This terminal focuses on the past, specifically the relationship the Sangheli and San’ Shyum had prior to the events of Halo 2… and how it was all a lie. The terminal also shows the remaining hostility between Elites and humans, but that both sides are willing to fight for a common goal, the defeat of a common enemy.
Bonus Terminal: The Battle of Earth
While the Terminal shows scenes of Epic space battles occurring (UNSC ships engaging Covenant ships, Orbital Defense Platforms being destroyed, etc.), all the command branches in the UNSC meet in order to be briefed by Keyes, Johnson, and the Arbiter about the events that occurred on Installation 05 and how the excavation site in New Mombasa are related. (Sets the stage for Halo 3)
All of these were educated guesses, if you feel something doesn’t fit or something is missing, feel free to voice your opinions.
Thank you for your time!