When Ralph Waldo Emerson uses the phrase "that divine idea each of us represent," he means the uniqueness of each person as conceived by God.

In his essay "Self-Reliance," Emerson urges all people to trust in their abilities and express themselves because he finds sanctity in the individual mind. He calls upon individuals to express their feelings and "that divine idea" with vigor rather than with diffidence:

The power which resides in him is new in nature, and none but he knows what that in which he can do, nor does he know until he has tried.

Urging each person to "[T]rust thyself," Emerson further encourages individuality because a rigid consistency in action and thought does not permit people to change and fulfill their potential. Since each person is an individual, Emerson wants his listeners to express themselves in strong words, even if what they say contradicts what they said the previous day. They just need to be true to themselves and trust themselves by applying their own standards--that "divine idea"--to what they think and do rather than following convention.

Emerson's "Self-Reliance" is a work which strongly promotes his philosophy of individualism.   Emerson, at the beginning of the essays, speaks of a time where every man comes to a halt in his own education and realizes nothing more can be bestowed upon him through traditional methods.   Humans are placed in a position where they can catch "one ray" if it were to fall, and that "we but half express ourselves, and are ashamed of that divine idea which each of us represent."  Emerson preaches not selfishness but the presence of divine spirit in every individual.   This concept embodies the idea of Transcendentalism:  each individual has potential for goodness/greatness and in a way, is his own mini-god.   Emerson, in the essay, discusses that self-trust is the means by which we discover "that divine idea which each of us represents."   Rather than fighting for new means, Emerson emphasizes for people to accept the position "divine providence has found for you, the society of your contemporories, the connection of events."   In other words, make greatness with one's current state of being.   

Compared to the early American thinkers, in particular, the Puritans, Emerson's views are highly radical/liberal.   Rather than having an elected few be in the company of the great spirit, all individuals have the potential to harness this direct power from God.    However, in order to do so, one must be independent aka "self-reliant."   

What is "that divine idea which each of us represents" in Emerson's "Self-Reliance"?

"Self Reliance" is largely about the importance of living true to one's own nature as a human being—hence the title of the essay. In Emerson's mind, to be truly human is to resist the will of what is popular, or is represented by tradition. It is not to act out of fear of censure but rather to embrace one's individuality and remain true to one's self.

The "divine idea" your question refers to is God as Emerson believes in him. At the same time, however, it also refers to God as reflected in the human soul. For Emerson, there is something divine in a human being, and this creates in each individual human the responsibility to live up to that intrinsic divine quality. Meeting that responsibility is the key criteria which defines if one's life has been well lived.

What is "that divine idea which each of us represents" in Emerson's "Self-Reliance"?

According to Emerson, the divine idea that each one of us represents is the role on earth that providence has selected us to fulfill. Each of us is born with a destiny implanted in us by God. We need to look into our souls to find that destiny and then trust in it. When we trust it, then our "heart vibrates to that iron string."

Great men, says Emerson, have always trusted in the divine idea of what they should be. We are, like them, to reject being cowards and instead march bravely forward to do our part in forwarding God's plan. This trust will lead to the betterment of all of humankind.

Instead of simply conforming to what our family members or our society tells us is the safe and secure path through life, we must seek to find what God means us to do and do it--even if it upset our family or our society.

What is "that divine idea which each of us represents" in Emerson's "Self-Reliance"?

Ralph Waldo Emerson's essay "Self-Reliance" is a major statement of the beliefs of the Transcendentalist movement.  One of these beliefs is that God is present in all things and that all things are connected together by the fact that they are all parts of what we tend to think of as God.

Therefore, in the quote you cite, the divine idea is God.  Each of us represents that divine idea because each of us is a part of God.

Because of this belief, Transcendentalists were often early advocates of women's rights and the abolition of slavery.