One of the first is that he immediately sees to the needs of his guest without asking questions, etc. It was important to the Achaeans that guests be taken care of, even when unannounced.
Another is his intense and long-suffering loyalty to his master Odysseus. One of the main themes of the story, particularly upon Odysseus' return, is the question of who has been loyal and who has forgotten him or attempted to move in on his wife and his lands after he has been gone for so long. Eumaeus has remained loyal to him despite the length of his absence.
A third might be his somewhat hidden wisdom, the fact that he can see right through Odysseus and picks his lies out from the stories he is telling immediately.