Most of the boss battles in Final Fantasy VII Remake's Hard Mode play out much the same as they do the first time around. Granted, they can often take a while longer due to enemies having higher HP, but generally, there's no need to alter the original strategy all that much. Sadly, this is not the case with Eligor.
On Hard Mode, several of his moves have been changed so that they now inflict negative status ailments on contact. He's also able to cast Reflect on the party; which can prevent players from healing if they go into the battle unprepared. As a result, a lot of players choose to throw caution to the wind and aim to take Eligor out long before he can ever really get going.
Setup
A lot of Eligor's attacks on Hard Mode can inflict Silence and Sleep, so you'll definitely want to pair orbs of Warding and Binding materia somewhere in Aerith's equipment to protect her from these ailments. Bring Cleansing materia too so that she can cast Resist on Cloud and Tifa. Revival and Healing materia are essential as always and you'll also want to equip her with Ice and Wind as Eligor is weak to these elements at different stages of the battle.
Another annoying thing about this fight is that on Hard Mode, Eligor will sometimes cast Reflect on the party which will prevent you from being able to heal. Equip Tifa with Subversion materia and pair it with Magnify to remove this with a single cast and bring an orb of Time materia to cast Haste on the party and Stop on Eligor. Due to Cloud's high magic stat, it's worth giving him Wind and Ice materia and pairing the latter with an orb of Elemental materia in his weapon. You can do the same for Tifa as well if you managed to get your hands on the second orb of Elemental materia.
When it comes to the remaining slots, use stat raising materia to boost everyone's HP and MP and give Aerith an orb or two of Magic Up. More cautious players may want to give Cloud and Tifa Revival and Healing materia as well, but you can get away without them if you're cautious or reckless enough. If you didn't steal Aerith's Bladed Staff from Eligor the first time around, now's a good time to do it and so make sure that somebody has Steal equipped. As far as summons go, Shiva is by far the most useful for this battle although you may not need her.
Weapon wise, you should give Cloud either the Buster Sword or the Twin Stinger and Aerith the Mythril Rod. If Tifa has Elemental materia equipped, use the Sonic Strikers, but if not you're better off going with her Metal Knuckles instead. The Gotterdammerung should go to Cloud if you have it, but otherwise, equip him and Tifa with either a Champion Belt or Supernatural Wristguards and give Aerith a Circlet.
Strategy
Phase 1
The best strategy for this battle is an aggressive one, so start by laying down an Arcane Ward with Aerith and then have her and Cloud hit Eligor with Blizzara a few times until he's pressured. Next, use Focused Strike and Focused Thrust and you'll stagger him very quickly. You should now be able to deal enough damage (around 30% of Eligor's total HP) to end the phase before he recovers, but if not, rinse and repeat until the phase is over.
Phase 2
Eligor will fly into the air at the beginning of the second phase and so use this opportunity to set a few things up. Have Tifa cast Haste on Aerith and use Aerith to cast Resist on Tifa and Cloud. Once Eligor lands, Tifa should cast Stop on him and then Haste on herself and Cloud. While this is all going on, Aerith can be casting Blizzard or Blizzara (don't forget to stand in the Arcane Ward) while Cloud either does the same or uses Punisher Mode to deal physical damage to the boss.
Once Eligor is staggered again, use your most powerful abilities to deal as much damage as possible with a view to taking his HP down to around 35%. If you're fast enough, you should be able to end the phase before he casts Reflect, but if not, have Tifa cast Breach on the party and then again on Eligor himself. If Eligor flies into the air for a prolonged period at any point during the phase, you can bring him down fairly quickly by hitting him with Aero a few times.
Phase 3
Eligor will start the third phase by using his Winds of Gehenna attack, so try to move your party members away from the center of the arena. You'll now be able to target his wheels as well as his main body, so cast Stop on one of them and then go to town on it until you cripple it. Take out the remaining wheel to stagger Eligor once more and then hit him with everything you've got (including Cloud's limit break if you have the Gotterdammerung equipped). Together with a few Blizzaga casts, this should be more than enough to finish Eligor off and end the fight.
Eligor's Key Attacks & Abilities
Dash 'n' Slash (Phases 2 & 3)
Eligor will swing his Javelin from left to right and anybody that he hits will be inflicted with Silence for 40 seconds. Keep out of the way; especially if you don't have Resist active.
Gallop (Phase 2)
While on the ground, Eligor will ride around the battle area crashing into anybody in his way. If he uses the move while airborne he'll instead hurl javelins at the party. You can block the javelins for some extra ATB charge, but try to avoid him while he's on the ground.
Javelin Bolts (Phases 2 & 3)
Javelins will fall from the sky like bolts of lightning and will deal area damage both on their initial impact and shortly after they land. You should be able to get a decent idea of where exactly they're going to strike, so dodge them by running and rolling out of their way.
Javelin Gallop (Phase 3)
Being hit by Eligor's javelin will inflict Silence for 40 seconds, so unless you've already cast Resist or have Warding and Binding materia equipped, you'll definitely want to steer clear of his swinging weapon.
Piercing Gaze (Phases 2 & 3)
This attack directly targets a single party member and will inflict Sleep and Silence for 15 seconds. In phase two you can hide behind the shipping container in the center of the arena to avoid the attack. Alternatively, if you have Resist active you won't be affected by the status ailments.
Reflect (Phase 2)
Unlike on normal mode where Eligor will only cast Reflect on himself, he will now cast it on the party as well. Cast Breach on the party as quickly as possible to remove it and then cast it on Eligor so that you can resume attacking with magic.
Scorpion Strikes (All Phases)
This attack will hit six times and can deal a fair amount of damage. Try to avoid it as best you can, but if you can't get out of the way, hold the block button to reduce the damage and build your ATB gauge a little in the process.
Stab & Swipe (All Phases)
These are very basic physical attacks that will inflict Silence for 40 seconds upon impact. Make sure that you have Resist active or dodge them by running or rolling.
Sweeping Gaze (Phases 1 & 3)
Eligor will shoot a beam across the battlefield which will inflict 15 seconds of Silence and Sleep if it makes contact. You should be able to dodge it, although if you have the Resist buff active you can probably afford to tank the hit.
Trample & Reaper (Phase 1)
These attacks aren't too damaging and can both be blocked in the event that you're unable to avoid them.
Winds of Gehenna (Phase 3)
This is a tornado-like attack that will suck in nearby party members before exploding. It deals quite a lot of damage, so make sure to spread your party members out and run away from it if you find yourself too close.