So, in essence, my friend is playing in my run of Waterdeep: Dragon Heist as an Aarakocra Arcane Trickster. Through some shenanigans, getting arrested, getting discharged into Varja's supervision at Blackstaff Tower, and roleplaying out his indentured servitude at Blackstaff Academy, he and I got to thinking that his PC might be a good fit for the school. He roleplayed well and convinced Varja to admit him- now I want to modify his progression as an Arcane Trickster to reflect the fact that he's spent some time studying under powerful wizards at one of the most prestigious schools in the Sword Coast.

My ideas so far are as such:

• His PC gets to borrow Arcane Recovery and the Spellbook mechanics from the Wizard class- in essence, nothing is changed about his level-ups and what spells he innately knows, but he can in theory learn any spell he finds copied in the world that he would be able to cast given his status as a half-caster, provided he has time to copy them and the gold to spend on materials.
• He picks a single school of magic to focus on, and get nerfed versions of the bonuses a Wizard committed to that school would (ex. if he committed himself to the School of Divination, he would only roll 1d20 for his Portent.) I know this might be potentially unfriendly to what an Arcane Trickster is supposed to be- a Rogue using magic to supplement their thievery- but his character development is also slowly pushing him to be a better person, and I'd like to help him reflect that with his powers.)
• He gets some form of "Mastery" benefits to his spellcasting (ex. he learns how to cast Minor Illusion without verbal components.)
• He gets spells that not even normal Wizards would get access to- nothing like Eldritch Blast or Smite, but something in the way of Vicious Mockery, where he learns how to emulate other class' abilities in a way an Arcane Trickster couldn't.
• I extend his level-ups beyond 4th level, where Arcane Tricksters usually get capped at (this would be good, as he's expressed interest in eventually using Power Word Kill- naturally, as a DM, I quake in my boots thinking about that, but my table's fun comes first.)

I'm not experienced enough with Wizards or Arcane Tricksters to know what either of them need in terms of balance, though I've read that Arcane Trickster doesn't scale very well into mid/high level as I intend to take this party to. Any help you can provide would be greatly appreciated- thanks!

Here's my suggestion for the simplest solution:

Give them the Ritual Caster feat. It gives them a spellbook that's not as versatile as a Wizard's spellbook (they can only scribe Ritual spells into it), but they learn at least two new spells and gain access to ritual casting (which an Arcane Trickster normally can't do). This would give them some unique, Wizardly functions in a way that's easy to integrate into an adventure and don't require any homebrewing on your part.

Feats are a great way to give players a non-tangible reward for doing something outside of what they would just get from leveling up more in their primary class. 

IMO, the best way to make an Arcane Trickster more “Arcane” is a 2 level dip into Wizard and take either the Bladesinger or Diviner as the subclass. With the green-flame blade cantrip the Arcane Trickster is pretty effing baller. Like, just letting them take a couple levels of Wiz doubles your cantrips, gives you ritual casting, gives you potential access to every 1st-level Wiz spell and gives the Trickster a decent amount of versatility with their 1st-level spells since they are ould get to prepare some of them from the Wiz levels, and all around doubles the “Arcane” half of the Trickster. Add on the subclass feature too and it’s on. And what you sacrifice from those two Rogue levels isn’t really that much of a loss compared to what you gain, it just delays stuff a little bit is all. Let them take 1 level of ASIs now and another one later if they want to to minimize that delay. That’s my suggestion at any rate. 

Ritual Caster or Magic initiate.

Arcane Trickster is pretty powerful if your not looking for a front line fighter. Its weak on spells but just adding ritual spells makes it way more useful. The wizard rituals spells are a helpful utility. Find familiar is a help for those sneak attacks from range in combat.

Wizard magic initiate adds a few cantrips and one first level spell outside of the Illusion and enchantment lists. Mage armor is a help if you do not want to use regular light armor.

As for cantrips I tend to go mage hand, minor illusion, and ranged attack spells. I like to try to cover as many damage types as possible with the attack cantrips That way I can do some type of damage in most cases.

As others have said, give them a free feat related to spellcasting for this RP. Feats are already balanced for the game so you are far less likely to break your game than if you try HBing something. Feats on this theme include:
Magic Initiate - some extra cantrips, and a 1st level spell of his choice
Ritual Caster - gives him the Wizard spellbook mechanics in a suitably reduced capacity
Fey/Shadow Touched - 2 free castings of a spell per day is pretty awesome for an Arcane Trickster considering their lack of spellslots
MetaMagic Adept - this lets him get Subtle Spell and use it twice per day for some shenanigans.

I would even think about removing the magic school restriction from the subclass altogether. 

I do not understand exactly why the AT even has a restriction. It must be a DD1 carry over from when everyone wanted to be a Thief/illusionist.

I think it was just to push the "trickster" part and not let the character start busting out evocations.

He's specifically said that he wants to focus in on Abjuration as a means of protecting his allies. My thought is that he wants to depart a bit from the usual role of an AT and delve more into becoming a utility caster.

At this point I may just tell him that he can reroll his character into a Wizard.

I’ll again say it’s just a fantastic, narrative, in-game reason to multiclass.

Quote from Roddu >>

He's specifically said that he wants to focus in on Abjuration as a means of protecting his allies. My thought is that he wants to depart a bit from the usual role of an AT and delve more into becoming a utility caster.

At this point I may just tell him that he can reroll his character into a Wizard.

 If that's what he wants he should swap to cleric, druid or bard.