The Greatest Day is a religion, founded by the Hosts. People in this religion, believes that the Hosts are here to help. It is based on The Hosts and The Arrival and attempts to merge all other religions into one.
Lindsey wears a lapel pin, that appears to be the same as the symbol for The Greatest Day, and Lindsey regularly attends church services along with other members of the Los Angeles Colony.
They also believe in "The Greatest Day" (which sounds similar to Alan Snyder's "Greatest Day In History") and believe that children will somehow have a big part to play in that day.