I have an npc in my game that is a third level abjurer. What kind of spell list would he have. 

Any he wants. But he should be focusing on having at least one protective spell at each level and preferably something you may want to cast more than once. The ward you get as part of your arcane tradition isn't super strong, the trick is to keep repairing it by casting abjuration spells so you can keep using it to protect yourself and your allies.

For 1st level spells Mage Armor and Shield are vital. Mage Armour is a good spell to get the ward activated and give you protection for a battle. The Shield spell will let you protect yourself for a round and your ward will regain 2 hp. You could also consider either Absorb Elements and Protection From Evil and Good. They are situational and you will get less out of Absorb Elements than a melee fighter but if you ever go against a dragon unleashing a breath attack the ability to halve the damage before you even make the save can save you life when combined with a ward.

There are no Abjuration spells for level 2 that will be useful for your ward but you can use the slots for shields if needed. However, since abjurer builds are support you can consider spells from other schools that will assist you and your allies like Blindness/Deafness - blind the enemy and not only will its attacks be at disadvantage but you and your allies can attack it at advantage, until it saves. Blur and Mirror Image are both great defense spells. Hold Person is decent as a control spell and situationally amazing but it does target Wisdom which will mean it can fail about half the time and even on success may not last long. You can consider Maximilian’s Earthen Grasp which is not as great but you can make more than one attempt and it targets Strength making if often more effective against spellcasters. Misty Step is useful, very useful and can even save you if you get surrounded by enemies: just bonus teleport 30 ft and use your move and dash action for up to 90 ft distance. Even if they dash they cannot get near you, letting you keep the distance and get breathing room. Invisibility can also get you out of danger and imposes disadvantage on attacks and help you with stealth if you need it. Great for getting away if the combat goes bad. Then, Suggestion, it's a weird, um, suggestion but this is incredibly useful. Being able to just tell an enemy to go to sleep, go run away and never come back, or just stand there while you set traps around them (suggestion to keep them standing there, somebody makes a Snare and somebody then shoves them into it and voila a pinata!).

For level 3 there's some good spells. Counterspell and Dispel Magic will be vital. The usefulness of these cannot be underestimated and at Level 10 these become more effective too. It cannot be said how clutch it is to counter a dangerous spell from an enemy or just instantly and safely remove a magic trap or an effect placed on an ally. Protection From Energy is also fantastic if going against elemental enemies.

Alarm is really good for patching up your Arcane Ward between fights since it's a ritual and you can just recast it as many times as you need. It's also pretty handy in any situation where you need to take a break inside a dungeon.