I mentioned that I have trouble playing kobolds and goblins differently. Two undifferentiated weak, trap-setting, underground cannon-fodder races is too many, so a while ago, I came up with some new quirks for goblins.
Here's my take on kobolds.
Kobolds are scavengers
I decided that goblins were producers - they don't need other races to survive.
Kobolds, on the other hand, don't produce anything. They live on the edges of settlements and steal garbage, set up traps for travelers, and raid farms. Kobolds are the first thing you fight at level 1 because they're right on the edge of town.
Therefore, kobold weapons are likely to be daggers, scythes, and shortbows created by other races. Their industry goes as far as weaving ropes into nets and turning leather into slings (but not making the rope or the leather).
When they're unable to steal food, kobolds eat bugs and small animals, and set traps to catch large animals and humans.
Kobolds worship fear
I had a bit of a problem coming up with a kobold gimmick. Recent editions have them as dragon-worshippers, which is fine, I guess, but it's limiting. For instance, you don't want to design a system that has kobolds and dragons (a high and low level encounter) always sitting on top of each other. What level party will face them?
In the past, I've also experimented with kobolds as demonic minions suitable for level-1 parties to face. They're scaly and have horns - all they need and pitchforks and the smell of brimstone to fill that role.
Kobolds' defining characteristic is that they're the weakest monster in the monster manual. Individually, they're afraid of everything. I decided that was the key to their psychology. They worship fear. If a creature is strong enough to kill a lot of them, they will serve that creature. The more frightened they are of the creature, the more fanatically loyal they become.
Kobolds will fear any creature that kills a lot of them, but the creatures they worship most slavishly are those with cause fear effects: typically dragons and demons have such powers. Conveniently, that preserves the lore of the dragon-worshipping kobold.
Kobolds like to cause fear
Kobolds try to inspire fear in others. They're too weak to do so with force of arms, so they do so by exploiting traps, darkness, and trickery. They prefer traps that don't kill instantly. Fiery logs rolling towards their target; poison that weakens over time; pit traps that hurt but don't kill; those are their favorite. They also have a weakness for Scooby Doo style trickery, terrifying villagers with phosphorescent scarecrows and devil masks.
When Mike Mornard ran us through a kobold maze, he had a handful of kobolds, striking from the darkness, terrifying a well-armed band of stronger first-level characters. That's part of a tradition going back to Gary Gygax's brutal level 1 kobolds. Kobolds are most effective when they're scary.
Like goblins, kobolds have their own brand of magic. Kobold magic lets them take on the characteristics of the thing they fear most. Thus, kobolds enslaved by a white dragon might gain icy dragon breath (for 1d4 damage). Kobolds enslaved by an evil necromancer might gain wimpy little Emperor Palpatine lightning. Kobolds used by drow might gain the ability to create shadows.
Kobolds will also gain obsessions related to those of their masters. A dragon's kobolds will hoard treasure. A vampire's kobolds will drink blood. An evil knight's kobolds might actually learn to march in step.
Some kobolds can't find any evil creature to act as their master/protector. These tribes tend to gravitate to the edge of human settlements, where they steal from, and to some degree imitate, adventurers and city guards.