In the Odyssey, she was the first person to recognize him after he returned home from the Trojan War, after he entered his own house as a guest of Penelope, disguised as a beggar. Euryclea bathed him and recognized him by a scar on his leg, which he had received from a boar while hunting with his grandfather Autolycus. Odysseus stopped her from telling Penelope or anyone else of his true identity.
Euryclea also informed Odysseus which of his servant girls had been unfaithful to Penelope during his absence, conspiring with Penelope's suitors and becoming their lovers. He hanged the twelve that Euryclea identified.
Later Euryclea helps Penelope confirm that Odysseus has indeed returned. Penelope tells her to move the bed Odysseus built in their marriage-chamber; Odysseus recognizes that it has been moved, and Penelope finally accepts him.
In other traditions Euryclea was also known as Antiphata.
Eurycleia is very kind and thoughtful and she is very quick to agree to wash the strangerÂ’s feet.
Eurycleia is also very civil with the suitors even thought she really hates them being in the house, but Melantho is quite “cozy” with the suitors and is on their side when it comes to them marrying Penelopeia.
Eurycleia had raised Odysseus since he was a child and she taught him what was wrong and what was right.
Not only that, but as the other servants were condescending towards "the begger" and Eurycleia was, indeed, the nurse who raised him, the King felt it was only appropriate for her to be the one to wash his feet.
Odysseus lets Eurycleia know also because she is as close to his mother as any living mortal women and knows that she is concerned for Odysseus' safety equal to or greater to the amount of Pennelope's worries.
Odysseus trusts Eurycleia because she had shown him the proper xenia, and he had known her eversince he was young.