Many people have interpreted this as a statement of either self-righteousness or competition -- as if we are somehow to compare ourselves to the world and come out on top. I would like to consider her words in a different light, because I think such interpretations may be misrepresenting her message.
First of all, in all but one of her comments that I could see, Sister Beck didn't say that we are the best. She said that we should be the best. I hear this as an invitation to consider who we are and what we are here to do.
I hear her call to us as being similar to what the Savior said to His followers:
Ye are the light of the world. A city that is set on an hill cannot be hid.
Neither do men light a candle, and put it under a bushel, but on a candlestick; and it giveth light unto all that are in the house.
Let your light so shine before men, that they may see your good works, and glorify your Father which is in heaven.
Matthew 5:14-16; see also 3 Nephi 12:14-16; 15:12)
We are a covenant people. As such, we are supposed to be different. The Lord expects something from us. We are asked by the Lord to be a light to the world. I think Sister Beck is calling to us in a similar way. At the end of her Relief Society broadcast talk, she said the following:
Our prophet, whom I sustain with all my heart, has said that there is a better way than the way of the world.
I think Sister Beck is trying to remind us of the need for us to rise above the philosophies and practices of the world. We should understand the importance of faith, family and relief better than anyone because we have the knowledge the restored gospel and continuing revelation give us. There are those outside our faith who are doing MUCH good in these regards, but they don't have the fullness of the gospel to frame what they do and believe. They don't know of temple covenants that bind families together. They may not fully understand the "doctrine of the family" to which Sister Beck refers. But we do (or we should), and it is on our shoulders to put that light on a candlestick and "be the very best" so that, among other things, others can know that there is a better way. There is more light and knowledge about the family than the world knows. How will they know if we don't live what we know?
Does this mean other people can't be or aren't excellent parents, loving Christians, deeply good and God-fearing people? Of course not. Does this mean that we reject the good that others do? Absolutely not. We rejoice in the good others outside the Church do. We stand together with good people all over the world to serve, to defend marriage and to strengthen the family, and to try to make the world a better place. We are deeply grateful for those who respond to the light of Christ in their lives and recognize the importance of God and family above all else.
But we are the only people in the world bound by priesthood covenants to do these things. We are the only group of people led by prophets called as mouthpieces for the Lord Himself. That gives us not a reason to boast, but rather a sobering responsibility.
For of him unto whom much is given much is required; and he who sins against the greater light shall receive the greater condemnation (D&C 82:3).
I hear Sister Beck expressing hope that we will rise to the duty that is ours to respond to the light that is ours by nature of the blessings of the Restoration. And seek to share that light in word and action. We should be living lives consistent with gospel principles better than anyone, not in a spirit of competition, but simply because we are the ones who know and have them. We can always do better, and I hear our leaders (all of them!) inviting us to do just that.