In what book of the Homer's Iliad did Athena first help for the Greeks?
Athena’s role in Homer’s Iliad actually starts before the text itself. She is one of the three goddesses who wish to be awarded the apple Eris throws to provoke disent among the wedding guests. Partly in anger over the apple being awarded to Aphrodite, she decides to support the Greeks.
Her first intervention in the Iliad occurs in Book I when she prevents Achilles from killing Agamemnon. In Book 2, as she continues to do throughout the Iliad and Odyssey, Athena gives good advice to Odysseus concerning how to manipulate the Achaeans.
Athena is the Greek goddess of war, defence, crafts, wisdom. Her role as the deity of war emphasizes her importance in the Trojan War. The best way to describe Athena's role is that of a seasoned veteran general, commanding the ranks with strategic brilliance and cool dispassion. This is in contrast to Ares, the god of war, who is depicted as a hothead young solider rushing into the heat of the battle. In general, the Greeks sought Athena's wisdom over Ares' tactics when they discussed war.
In the Iliad, it is important to note that Athena sides with the Greeks because of the Trojan prince Paris' original snub (when he awarded the golden apple to Aphrodite instead of Athena). The other reason she sides with the Greeks is because of her affinity and affection for the Greek hero Odysseus. They are "kindred spirits" in a way because they both prize cleverness and brains over brawn--the reason why they both succeed in various pursuits.
She first appears in the Iliad in Book 1 to warn Achilles against feeling too much rage against Agamemnon. She wants the Greeks to win, and knows that Achilles' anger will be a huge detriment. She promises him gifts if he attempts to hold back his rage, essentially trying to bribe him.