The idea behind "offering it up"--taking the moments and events, the sorrows and the joys, the sufferings and the consolations of each day--is Eucharistic. It's the way that we live the Eucharist in our daily lives.
As I was preparing my talk for last weekend's Sacred Heart Communities in Collaboration conference in Willimantic, CT, I ran across some words of Pope Benedict that confirm this. Here's what he said to a conference that was held by the Diocese of Rome last week:
"The celebration of the Eucharist challenges us and at the same time empowers us to become, in our turn, bread broken for our brothers and sisters, coming to meet their needs and giving ourselves. Because of this, a Eucharistic celebration that does not lead us to meet people where they live, work, and suffer, to bring them the love of God, does not show the love it contains. To be faithful to the mystery that is celebrated on the altars we must, as the Apostle Paul exhorts us, offer our bodies, ourselves, as a spiritual sacrifice pleasing to God (Romans 12: 1) in those situations that require dying to our "I" and which constitute our daily 'altar.'"