For Thane’s loyalty mission, you will need to help the talented assassin to stop his son from making the same mistakes in life that he did. The mission takes place upon the Citadel, so travel there after speaking with Thane and getting the mission.
Upon arrival, head for the C-Sec Offices beside the docking area and speak with Captain Bailey. Ask about Thane’s son and Bailey will tell you where to go. Make your way to Zakera Ward Level 28 and then go to the Comm Terminal area. Here you will find Mouse, who will tell you what you need to do. After that, go back to Captain Bailey who will organize an interrogation of Elias Kelham.
At this point you will need to decide if you are the good cop or the bad cop. You should pick the one that suits your morality on the play through, so Paragons should be good cops, and Renegades should be bad cops. This will lead you to the information you need, and the next portion of the mission.
For the next section you will need to follow a target , keeping them in sight and updating Thane on their position. If you lose sight of the person, this will count as a failure, and Thane will not become loyal. There is nothing wrong with a little bit of save-scumming to get through this part.
Just follow the Turian, updating Thane on his position every chance you get. Just keep following the Turian until you come to a moment where you need to use either a Renegade or Paragon action to stop Thane’s son from killing him. This is another point where you can fail the mission overall, and lose Thane’s loyalty.