Why does Sir Gawain feel he is the one best qualified to accept the Green Knight's challenge in Sir Gawain and the Green Knight?
In Sir Gawain and the Green Knight, Sir Gawain's acceptance of the Green Knight's challenge isn't based so much on his feelings of being the best qualified but rather on factors akin to default. None of the knights in King Arthur's court wants to accept the Green Knight's morbid Christmas game; it spells certain death for the player. Under the duress of needing to uphold the strict chivalric codes of honor, King Arthur himself agrees to be the participant.
Even when King Arthur steps forward, all the knights continue to remain silent, except Gawain. Gawain speaks up and says that the King should be spared and that he himself should be allowed to take the King's place because of his stature as a knight of insignificance.
Sir Gawain's acceptance of the challenge then is by default, as it were, because the other team, the more able and less lowly knights, didn't "show up" to accept the Green Knight's challenge. In summary, Gawain's greatest boast of quantification for the challenge is humility.