On this page of our guide to TES V: Skyrim you can find a detailed description of Bring a Follower to Delphine side quest (The Blades faction quest). It's a unique task, because you have to recruit new members for The Blades.
You can approach this quest no sooner than after finishing the main quest Alduin's Wall. Search the Sky Haven Temple for Delphine and ask her if the Blades are currently formed by only two people, namely her and Esbern (screen above). She will confirm this fact and ask you to recruit new members.
The game won't add any markers, as only people who you choose yourself will be able to join the Blades. After finding a proper candidate, return to Delphine and tell her of bringing a volunteer (screen above). You don't have to worry about losing his for good, as you still will be able to ask him to join you on your adventures. What's more, he will also undergo training and receive the Blades equipment. Notice - Only warriors can join the Blades. Lydia seems to be the perfect first candidate. In search of more volunteers, you should definitely head to Jorrvaskr in Whiterun, of course assuming you have already finished the Companion quests.
You will need to bring three different characters to Delphine in total, as only then will the Blades gain enough power and you will be able to move to the next mission - (The Blades) Ask Esbern about dragon lairs.