The simplest way to increase a mount's AC is to use the same method we use with PCs: Armor (technically, Barding).
Barding is armor designed to protect an animal’s head, neck, chest, and body. Any type of armor can be purchased as barding. The cost is four times the equivalent armor made for humanoids, and it weighs twice as much.
If you have the money to pay for it (and DM approval to find a blacksmith that will make or sell it to you) you can get any kind of armor made for your Griffon mount, up to plate mail for 18 AC (though at 6000 gp, that might be a stretch at first).
There is also the question of whether your mount is proficient with Armor, and, if so, what kinds? The question "Are mounts proficient in armour (barding)?" looks into it and argues that, while the rules don't specifically talk about mounts having armor proficiency, any war-trained mount should be considered to be proficient with armor.
The fact that you're summoning your griffon from (I assume) a Figurine of Wondrous Power might present a complication with this. Before spending money on barding, you should check with your DM as to whether armor you put on your griffon while in active form will persist each time you animate it. I see no issue with allowing this, but you should check with your DM first.
The Mounted Combatant Feat
Using the Mounted Combatant feat will help your mount not get attacked, although it doesn't directly increase AC. One of its features is:
You can force an attack targeted at your mount to target you instead.
With this feat, you can ignore your Griffon's low AC by forcing attacks to target your own, possibly higher AC. With Mage Armor and 20 DEX, you could have 18 AC, which would functionally apply to your mount as well as yourself. (That you now have to watch your HP pool more carefully is an included risk of this approach.)
Another advantage of this approach is that some attacks will do reduced or no damage against your mount:
If your mount is subjected to an effect that allows it to make a
Dexterity saving throw to take only half damage, it instead takes no
damage if it succeeds on the saving throw, and only half damage if it
If you plan on using your mount a lot, then when you get to level 4 this feat may be a good way to preserve your mount, with the cost of using your ASI for a feat instead of increasing your primary spellcasting ability.
As you mention in your question, you can use Mage Armor to increase AC to 15 (assuming a normal 15 DEX griffon), but this will not stack with actual armor and cannot be enhanced:
You touch a willing creature who isn't wearing armor, and a protective magical force surrounds it until the spell ends. The target's base AC becomes 13 + its Dexterity modifier. The spell ends if the target dons armor or if you dismiss the spell as an action.
Normally, if one is not wearing armor, their AC is 10 + DEX modifier; with Mage Armor on, it is 13 + DEX modifier. So, regardless of your own stats, casting Mage Armor on your Griffon would give your mount 15 AC: 13 + 2 (the griffon's DEX mod).