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When you talk to Amir outside the gates of Barcelona, you can ask to become a Favored One of the Order of Saladin (aka the Knights of Saladin). Since that is a step (or three) short of actually joining the order, it is only considered a minor faction, and completing quests for it won't fulfill DaVinci's quest to join a faction.

Here are the quests you must complete to become a Favored One:

1. Find Ali Huban - If you talk to the twin brothers Farshid and Farshad (the dueling pair right next to Amir), they'll tell you that Ali Huban is a master of disguise and that he can probably be found in the Port District. They're right. You'll find Ali Huban in the Port District Tavern, disguised as Consuela the Barmaid. Ask Consuela for information, then about Ali Huban, and finally about news for Amir, and Consuela will admit who she is (100 experience) and give you a letter for Amir. When you give the letter to Amir (250 experience, 65 karma), you'll complete the quest.

2. Recover the Shard of Dreams from the Slavers - The slavers can be found in the Slave Pits far to the west and then south of Barcelona. When you confiscate the shard from them and return it to Amir, you'll gain 1200 experience and 65 karma.

3. Pass the Test of the Dream Djinn - In order to meet the djinn, you'll first have to donate a gem (150 experience). You can get a gem in a few places: from the water dryad once you've completed her quest, from Gomes in the Port District, and from Weng Choi. Then when you meet the djinn, you'll be given a choice of selecting either a physical or mental challenge.

Physical: You'll have to face four waves of enemies, but once you do you'll receive the Dervish of the Crescent perk (+5 to all combat skills) plus the Sacred Scimitar (2-11+3 damage, +5 critical chance, +10 pierce resistance).

Mental: You'll have to answer three riddles correctly. Since you can save before meeting the djinn, and since the riddles are multiple choice, you should have a relatively easy time with the challenge. Once you answer the riddles correctly, you'll receive the Scholar of the Crescent perk (+1 intelligence, +10 speech).

The Dream Djinn quest is the last one. Once you complete it and return to Amir (2000 experience) you'll become a Favored One of the Order of Saladin.