A Lashtail Hatchling

How to obtain the Lashtail Hatchling companion as Horde

To see the Lashtail Hatchling inside the Zul'Gurub instance after killing Bloodlord Mandokir you must first complete a series of small quest lines in Northern Stranglethorn.

*SPOILERS Start at step 7 for Horde - You can stop right before those steps if you want to see the main meat of the story unfold yourself*

1. 
   
2. As a member of the Horde this entire ordeal starts with The Defense of Grom'gol: Raptor Risk given by Commander Aggro'gosh in Grom'gol.
3. 
   
   
4. After completing that quest, you will automatically receive the quest A Lashtail Hatchling and the Lashtail Raptor Egg Fragment.
   Return with the hatchling to Far Seer Mok'thardin in Grom'gol (If your bags were full check your Mail for the Egg Fragment).
5. 
   
   
6. Next he will send you on a quest called I Think She's Hungry to feed the baby raptor murloc meat. Make sure your raptor is summoned so she can take bites out of the corpses as you kill them.
7. 
   
   
8. Now take the hatchling to the Bal'lal Ruins @ 33.7, 37.2 where you will find a pile of skulls on the ground, Your hatchling will run up to the pile and grab a skull from it. Now he will give you the quest Favored Skull which needs you to return to Nimboya in Grom'gol.
9. 
   
   
10. The cauldron near Nimboya will now offer the quest A Nose for This Sort of Thing.
11. 
    
    
12. Next you get the quest Who's a Big Troll? then Skullsplitter Mojo
13. 
    
    
14. After completing the above quest line, Bloodlord Mandokir will spawn and offer a "quest" of the same name. He really just wants to steal the raptor from you because it's special. Do not chase him when he runs off. There is more to do, you must now embark on a chain to try and save your new raptor friend from her fate.
    
15. 
    
16. Nimboya will now give you a quest to speak to Priestess Hu'rala.
17. 
    
    
18. She will lead you on a chain to try and save your new raptor hatchling friend from her terrible fate of becoming the Bloodlord's new raptor slave. This is all your fault! But maybe you can do something about it...
    1) Mind Vision, 2) See Raptor, 3) Mind Control, 4) Be Raptor, 5) Surkhan
19. 
    
    
20. After that really cool and fun quest chain from the Priestess you are ready to enter the 85 Heroic Dungeon of Zul'Gurub and face the Bloodlord Mandokir. It's time to undo the mistake you made by bringing him back to life, and you must save your new friend from a world of torture and evil. Once the Bloodlord falls at your hands, you will see An Old Friend.
21. 
    

How to obtain the Lashtail Hatchling companion as Alliance (Credit to user: Madwolf)

*SPOILERS Start at step 8 for Alliance - You can stop right before those steps if you want to see the main meat of the story unfold yourself*

1. 
   
2. For Alliance this chain starts with the quest Bad Medicine given by Brother Nimetz at the Rebel Camp.
3. 
   
   
4. Upon completion of Bad Medicine you will automatically obtain the quest Just Hatched and the Lashtail Raptor Egg Fragment. Return to Corporal Sethman at the Rebel Camp with the new hatchling.
5. 
   
   
6. He will then send you on the quest I Think She's Hungry to feed the baby raptor some basilisk meat. Make sure you have the hatchling summoned while killing basilisks so she may take bites from the corpses as you kill them.
7. 
   
   
8. Next you must do The Fate of Kurzen given by Lieutenant Doren in the Rebel Camp. Continue on his chain and do Spared from Madness next.
9. 
   
   At this point you must log out and in to spawn an NPC that will be required in the next step.
   
   
10. Now take the hatchling to the Bal'lal Ruins @ 33.7, 37.2 where you will find a pile of skulls on the ground. Your hatchling will run up to the pile and grab a skull from it. Now he will give you the quest Favored Skull. Return to Osborn Obnoticus at the Rebel Camp (This is the NPC that you must have logged out then back in to see according to some Alliance players).
11. 
    
    
12. Osborn will then give you A Nose for This Sort of Thing to start trying to figure out who that skull belonged to.
13. 
    
    
14. After that do the next quest in the chain, A Physical Specimen.
15. 
    
    
16. After completing the above quest line, Bloodlord Mandokir will spawn and offer a "quest" of the same name. He really just wants to steal the raptor from you because it's special. Do not chase him when he runs off. There is more to do, you must now embark on a chain to try and save your new raptor friend from her fate.
17. 
    
    
18. Osborn will give you a quest to go talk to Priestess Thaalia at Fort Livingston.
19. 
    
    
20. She will lead you on a chain to try and save your new raptor hatchling friend from her terrible fate of becoming the Bloodlord's new raptor slave. This is all your fault! But maybe you can do something about it...
    1) Mind Vision, 2) See Raptor, 3) Mind Control, 4) Be Raptor, 5) Ghaliri
21. 
    
    
22. After that really cool and fun quest chain from the Priestess you are ready to enter the 85 Heroic Dungeon of Zul'Gurub and face the Bloodlord Mandokir. It's time to undo the mistake you made by bringing him back to life, and you must save your new friend from a world of torture and evil. Once the Bloodlord falls at your hands, you will see An Old Friend.
23. 
    

From what I can figure out, the quest chain has you losing the little hatching, only to be reunited with the little bugger after doing Zul'Gurub on heroic (lvl 85). This means you won't have the little guy as a companion while you are lvl 29 and finishing up your Stranglethorn quests; instead you'll get him back once you reach lvl 85 and do the dungeon.

I, personally, think it is a cool concept that results in tying your early adventuring days to your later end-game adventures.

Just a note, if your bags are full, don't worry, return to grom'gol and you can get the quest from far seer mok'thardin.