What epic theme does the first part of The Lord of the Rings and The Odyssey share?
One shared theme that is present within both works is the idea of how the journey molds and shapes protagonists. Frodo's journey with the ring is one filled with challenge and adversity. Odysseus' journey back to Ithaca is much the same. Through the obstacles both face in their respective journeys, more is learned about themselves within themselves. The protagonists change and undergo a new level of self- reflection as a result of their voyages. Additionally, there is significant change in both characters because of their journeys. The idea of undergoing significant alteration to emotional and moral consciousnesses of Frodo and Odysseus as a result of their journeys is quite a valid one.
I suppose you are after something like "The fate of many is sometimes determined by the choices of one or a few."
In both stories the protagonist has great responsibility since the consequences of his acts are far-reaching. Because of this, leadership is portrayed more as a responsibility and even a burden rather than a privilege: "To whom much is given much is required."
In both there is also the idea of a quest (something to be found or a challenge to be met), which is a leit motif often appearing in the epic genre.
Other common elements of the epic are a long journey or war and the story covering several generations (a saga).