Perseverance basically means continuing on with a task or duty even if the going gets tough. And if there is one virtue Tolkien seems particularly to admire, it's perseverance. He loves characters who work their butts off even when they really don't want to continue. This might describe the entire Fellowship, in fact: the Company keeps going across rough terrain because they know they have to, to continue the Ring quest. But, if you think about it, the bad side also perseveres: in its hunt for Frodo. In the end, the characters' perseverance against the odds makes The Fellowship of the Ring a suspenseful read.
The ages of the characters in The Fellowship of the Ring affects their level of perseverance. Younger characters are more likely to want to give up.
If Frodo decided to give up, the rest of the Fellowship would support his choice.