We should distinguish between "the target(s)" of a spell that the caster has to pick when targeting the spell (where line-of-effect comes into play), vs. the creatures in the area of effect which are its "targets". 5e rules do use "target" in both senses; it's not exclusively used as a technical term with the spell-targeting meaning. Even the Fireball description itself uses "target" where many other AoEs (like Shatter and Lightning Bolt) say "a creature" (in the area). For some spells these are the same, like "up to n creatures within range of your choice" but for like Fireball it's different.
See also:
- What counts as a target for a spell? - more discussion of the distinction between the spell-targeting meaning of "target" vs. other usages in 5e.
- @Thomas Markov's answer to this question (disregarding the quote from the rules about targeting yourself; that's for spells that target creatures within range, not points or areas.) I think Thomas's answer is arguing for the same conclusion I am, that only one creature outside the AoE's area can be targeted by a morkoth with its spell-reflection ability, not other creatures around that target, but that isn't explicit in the answer.
- My answer about an ability that requires a goblin to "targeted" by a creature, unlike the Morkoth ability which only requires saving against a spell and picking one new target for it. In that other case, the distinction between "target of the spell/ability" in general vs. the target(s) picked by the caster/user was one of the reasons an ability couldn't be redirected at all. (It was phrased that was because it only works on attacks, but the "by a creature" distinction was interesting, and relevant to this question.)
The Morkoth can make one creature take the fire damage instead, but not change the point-within-range location of the AoE
The fireball still engulfs the area within 20ft of the point chosen by the caster, (potentially) damaging every other creature in the area, regardless of what the morkoth does. The morkoth can't change that targeting choice, only the fact that some of the spell's energy was coming at it. That's what being "a target" of an AoE spell means.
And no other creatures around the new target are affected (if they weren't already in the area); it's just reflecting part of the fireball's energy at one creature1.
(It's up to the DM to flavour that for narrative purposes into something that makes sense, like whether the reflection target is outlined in fire, or if the heat energy just goes into their body. And how it makes sense that they still attempt a dex save for half damage, or for none with evasion. Note that it's fire not thunder damage, so it's heat not explosive force that's doing the damage.)
In general, Spell Reflection can retarget a spell's effects only to the extent that the morkoth would have been affected by (was a "target") in the first place. e.g. reflect one ray of Scorching Ray with its one reaction, or the part of an upcast (multi-target) Hold Monster or Charm Monster where it was one of the targets. (The new target would then be charmed by the original caster, not the morkoth, I think, but that's a separate question from AoEs. Still, it's a useful thought experiment in terms of thinking of it as just modifying the targeting or transferring the effects an existing cast of the spell, not recasting the whole spell at a different "target" in the spell-targeting sense.)
Chain Lightning could get narratively weird for this reading of the rules, especially if the morkoth was the primary target that 3 other bolts were going to fork off of. Perhaps if the morkoth was the primary target, you'd apply the spell's limit that the secondary targets must be within 30 ft of the primary, even after spell reflection changed that target. But Chain Lightning does target creatures directly, so it's a separate question from AoEs.
Being a target of a spell doesn't necessarily mean being the target of the caster directly
When casting Fireball, the caster targets a point within range (150 ft), creating a magical effect there. That magical effect in turn targets all creatures within 20 ft, making them each save vs. fire damage.
Creatures in the area of an AoE are targets only in the general sense (of the spell (effect)), not the spell-targeting sense (targeted by the caster). That's why they can be outside the 150 ft Range of the Fireball spell, and why Sanctuary doesn't require a Wis save to target a point near a warded creature.
The Fireball description actually does refer to each creatures in the area as "a target" (of the fire created by the spell).
A bright streak flashes from your pointing finger to a point you choose within range and then blossoms with a low roar into an explosion of flame. Each creature in a 20-foot-radius sphere centered on that point must make a Dexterity saving throw. A target takes 8d6 fire damage on a failed save, or half as much damage on a successful one.
It's this sense in which a morkoth can make another creature take its place as a target. The morkoth can replace itself with one other creature (within 120 ft) as one of those targets of the fire damage (save for half), but that's all.
For that one target of the spell, it replaces the normal rule for determining affected creatures ("within 20 ft of the point" for Fireball) with a requirement to be within 120 ft of the morkoth.
Many AoE spells don't mention the word target, e.g. Lightning Bolt2 (range: Self (100-foot line)) has very similar language for the end of its first paragraph, except using "a creature" instead of "a target" in the last sentence. Similarly, Shatter just says "a creature" both times instead of "a target" but is otherwise worded very much like Fireball. So lets pretend the question used Shatter instead of Fireball as an example of not mentioning targets. IDK if Fireball's wording is meant to include objects as targets that don't get to make saves, or if that's just a lack of standardization of wording when they meant to say the same thing.
For spells that don't use the word "target", I think we should follow Fireball's example and consider any creature in the area to be a target of the spell (effect), despite not being targeted by the caster. @Thomas Markov's answer makes some good arguments that affected creatures should be considered targets of the spell.
Footnote 1:
If the morkoth's target was already in the fireball's area, could it take fireball damage twice? That seems narratively fine, like a reflector concentrating the energy, despite the fact that casting fireball in an enclosed space doesn't RAW do that. Or that covering a huge or gargantuan creature with a fireball instead of just getting one leg doesn't do any more damage.
In game-balance terms, a new ability was used, costing the morkoth's reaction.
But mechanically, that does mean the creature would be targeted twice by the same spell; the morkoth can only make it a target of the original spell when it uses Spell Reflections, not the target of a separate harmful ability that transfers the damage or effect.
So I think if a creature was already in the fireball's area, RAW a morkoth couldn't make it take damage twice from the same fireball. The situation would probably only ever came up in a battle with 3 or more factions, or a dominate monster on a morkoth. Or from accidental friendly fire.
Combined with careful spell metamagic, would the target still have advantage on the save if the caster had been careful of them but not the morkoth? Probably not; the spell energy reflected by the morkoth not reduced by careful spell, so now the target feels the full brunt of the spell. But RAW doesn't have much to say about this corner case; various possible rulings are compatible with RAW. This one is based as much on my narrative interpretation as anything else.
Footnote 2:
Historically, some DMs like to have Lightning Bolts reflect off walls in their games. At least Matt Mercer has said so on Critical Role, and used that homebrew there.
So for LB specifically, there's some historical precedent for having the line itself reflect in some cases, changing the shape of the spell other than cutting it short. Morkoth Spell Reflection seems perfectly suited to do that if that's what the DM wants. But that's 100% homebrew territory as far as 5e RAW is concerned.