Thane has a fairly quick mission and it's another non-combat one. You need to go to the Citadel for this one. Head on inside and talk to Captain Bailey. He's the one you talk to after security flags you on your first entrance, so you should know who he is.

click to enlarge
You can ask him about Thane's Son and also get information on Garrus' mission if you have it. Thane's son apparently was seen talking to a few known criminals. We need to go find Mouse. He's up by the Expedition store just one floor above the security section. He's also talking to someone on the phone when you approach, so he should be easy to spot.
Talk to Mouse and make your conversation choices. You can use a renegade interrupt to rough him up and get the information. Being nice will work too though, since he knows Thane already. He'll give you a tip on who the contact was. Since you've got a name, go back to Captain Bailey and ask him about your new. He'll begrudgingly bring in Elias Kelham for an interrogation. You actually shouldn't be able to blow the investigation. He'll talk eventually.
Someone with a lot of renegade can successfully convince him that they're a spectre, which will get him to talk immediately. You can also punch him three times through the renegade interrupts to get the information. This has no apparent consequences. You can charm him too, if you're committed to being a paragon.
It seems that Thane is going to kill an anti-human politician called Joram Talid.