So you're looking to tame a blue dragonhawk, but none of your friends will actually come with you to the Sunwell. Have no fear, for this beauty is absolutely solo-tamable at level 80!
Preparation
1. Absolutely bring at least one corehound with you. If you do everything right, you'll only need the one, but it's very nice to have backups. If you intend to keep your corehound, then make absolutely sure you have at least one active pet slot open.
2. Bring haste potions.
3. Stack as much haste and stamina as you can. For me, this meant wearing all PvE gear, changing out my trinkets for the Brewfest ones. I also put up a blessing of forgotten kings and nibbled on one of my spare fish feasts. (This may not be strictly necessary, but it's nice to have.) If you're an engineer, rock the haste enchant on your gloves.
4. Find a random person to group with you in a raid for a few minutes.
The Tame
1. Wait for the mechanical patrol to circle past the entrance. If you're looking into the front courtyard from the entrance, the patrol should be around the 6 or 7 o'clock position when you slip past to get near the closest trash pack. You could always kill it, but this way worked for me.
2. Get your corehound out and activate ancient hysteria. It lasts for 40s, so you have lots of time to work with, but you'll still want to move fast.
3. Abandon or dismiss your corehound.
4. If you feel daring (I didn't), launch an ice trap (or frost grenade, if you're an engineer) into the trash pack ahead of you, then very quickly....
4. Pop your haste potion and deterrence, plus any other haste buffs you have (engineering glove enchant, herbalist lifeblood, troll berserking, etc.). Tame beast and pray.
5. You will almost certainly die here. I didn't even see my tame complete in order to get a feign off. Even if I had, the trash would have just killed my dragonhawk and then re-aggroed me when I stood up.
6. Res and check your active pets to see if your dragonhawk is there. If not, you won't have an insanity debuff, so you can try again in a few minutes. If so, then congratulations on your new dragonhawk!

Thanks to Sigrah for the tips on ice trap usage and the engineering glove enchant, and Acherontia for the frost grenade suggestion!