What god assists Odysseus in his battle? In what ways?
The answer to this can be found in Book 22. The god (or goddess) is Athena.
Athena comes to Odysseus and scolds him for not being more aggressive. She really wants him to kill all the suitors. She turns into a swallow and flies up into the rafters of the hall. From there, she sort of controls things.
She makes the spears that the suitors throw all go astray even as Odysseus and his allies can throw straight. The other thing that she does is to make the suitors get all panicked. They become so overcome that they just run over to the other side of the hall and cower there while Odysseus and his men kill them all.
I think you are referencing both Zeus and Athena. Being the goddess of both wisdom and war, Athena gave Odysseus the wisdom to create an intricate plan which would assist four men in killing well over one hundred. Ultimately, Athena also gave Odysseus and his three faithful the cunning and stamina war required.
Zeus, all-powerful, likewise sat on the side of Odysseus giving his support. Not only is Zeus powerful enough to aid by giving power or strength, but he could also take away or make difficult the way of the suitors.