What is the lesson in morality in Beowulf?

Beowulf is an epic that came out of a
warrior culture and was used not solely to entertain but also to teach the values of
this culture.  The Anglo-Saxons valued bravery, heroism, strength, loyalty, fame through
glorious deeds, and the desire to fight evil.  Their leaders were their best warriors. 
Beowulf himself embodies all of these qualities, and you can find examples of these
throughout the text.  He is the strongest and best warrior in his country, and after he
defeats Grendel and his mother (among other victories) he becomes king.  Wiglaf, who
becomes Beowulf's successor, also embodies these traits.  When all of the other soldiers
abandon Beowulf in his fatal battle against the dragon, Wiglaf does not.  He displays
courage and loyalty in helping Beowulf defeat the dragon, which is why he is named the
next leader.  It is true that Beowulf fights these monsters to help his tribe and the
Danes, but he also does so for fame and glory and hopes to be remembered for such deeds,
which is why, on his deathbed, he requests that a monument be erected to remember him
and how great he was.  In seeing this monument, future generations will remember Beowulf
and the qualities he embodied, reminding them to fight evil forces with courage and
strength and to always be loyal.  Anglo Saxon culture did not value
humility.

Another interesting aspect to note is the idea of
Christianity.  By the time Beowulf became a written text and not
just an oral story, Christianity had begun to spread through Britain.  As such, certain
Christian ideals and language are sprinkled throughout the text we read.  The most
significant Christian value in this text is the connection between God and good.  When
battles against evil are won, the characters praise God and rely on him to help. 
However, the idea of faith is at odds with the pagan concept of fate, which is often
associated with unfavorable conditions and results in the
text.

So, to simply answer your question, a moral person in
Beowulf is one who bravely and courageously fights evil for the
sake of his people (and others) and for fame, someone who strives to be strong and is
loyal to his leaders/followers; increasingly, over time, this morality also began to
include a faith in God's help to win battles against evil.