One my favorites from myth is Theseus.  Unlike a Hercules who is little brain and at times too much brawn, Theseus is both a thinker and a fighter.  Although, as is almost always the case, there are some darker issues in the stories about him, it is not accidental that Theseus was considered the Athenian hero par excellence.  Take a look at Edith Hamilton's Mythology for a brief and well-written retelling of the major Greek myths.

Of course, one cannot forget that great figure of duty over desire: pious Aeneas.

If you're interested in some variations on the concept and tradition of a hero, you could also look into the Byronic hero (Romantic) and the anti-hero (modern; T. S. Eliot's J. Alfred Prufrock is a good example). 

 My favorite literary hero is Odysseus, protagonist of Homer's Odyssey,

the central male character in a novel, play, poem, etc., with whom the reader or audience is supposed to sympathize

The legend of the Odyssey centers on Odysseus' twenty year journey home from the Trojan War. He encounters fantastical monsters, must overcome temptations, and faces the trials and tribulation of the Gods. He is admired for his fearless leadership (fighting through the terror of the Cyclops), and although makes human errors (giving in to the lure of Circe), he is noble in his pursuits (knowing he must return to his wife and family).  The audience sympathizes with his situation, a man separated from his wife and sons, fighting his way with remarkable strength and heroism, facing danger and leading his men with relentless courage. Odysseus is "admired for qualities or achievements and regarded as an ideal or model" not only by his followers, but by his enemies who face his wrath and revenge in fear. The reader roots for this "toy" of the gods, and celebrates his revenge upon his return.

This is going to bring out a great deal of responses.  I would say that Classical heroes such as Hector, who demonstrated great skill and greatness on the battlefield with a sense of loyalty to both family and his citizens.  I would say that Perseus is another great hero for his strength, ability to think critically out of arduous situations, and for his leadership qualities.  In these settings, heroic qualities and traits are external, ones that are tangible and evident for all to see.  A more subjective example of a hero would be Stephen Daedalus from Joyce's "A Portrait of the Artist as a Young Man."  I find Stephen to be heroic because he is willing to embrace change and find his own true voice through multiple epiphanies and without fear of change.