Rather than believing himself to be worthy or the best qualified to accept the Green Knight's game, Gawain is the one to take the Green Knight up on his challenge simply because he wishes to spare King Arthur from having to do it himself. In fact, the text supports the idea that Gawain believes himself to be lesser qualified than many of the knights in attendance; his is simply more ready than they are to protect the king from what would have been seen as certain demise.

Upon the Green Knight's entrance into the hall during the feast and explanation of his reason for coming, the onlookers, including Arthur's knights, are stunned into silence. Indeed, it is the silence that causes Arthur to be the one to accept the Green Knight to protect his personal reputation as well as the reputation of the court. It is only once Arthur accepts that Gawain steps in and states,

I am the weakest, I know, and of wit feeblest. / Least worth the loss of my life, who’d learn the truth. / Only inasmuch as you are my uncle, am I praised.

Thus, the humble and loyal Gawain is the one to trade blows with the Green Knight because of his willingness, not his worth.

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.

Why does Sir Gawain consider himself the knight best qualified?

When the Green Knight issues his challenge, everybody in King Arthur's court at Camelot hangs back. The Green Knight begins to mock the court, causing Arthur himself to step forward, ax in hand.

At this point, Sir Gawain intervenes, offering himself instead. While we might expect him to base his claim on his great strength or courage, he instead argues it on the basis of his unworthiness, saying his life is the least valuable of any. He states he is the "weakest" and has the "feeblest" wit. He also notes that his honor comes not from his own accomplishments but from his status as the king's nephew. He also says that he should be allowed to stand in for Arthur because he was the first to ask. Gawain says, too, that this "folly" should not fall on the king:

I am the weakest, I am aware, and in wit feeblest,

and the least lost, if I live not, if one would learn the truth.
Only because you are my uncle is honor given me:

save your blood in my body I boast of no virtue;

and since this affair is foolish that it nowise befits you,

and I have requested it first, accord it then to me!

As events will show, this is false humility on Sir Gawain's part. He actually believes he is a chivalrous and worthy knight. Later, when his life is on the line and he is tested, he behaves dishonorably, withholding the truth of the green girdle from his host. It is only after his adventures with the Green Knight that his limitations as a person become real to Gawain, who returns to Camelot chastened and wiser.

Why does the Green Knight challenge Sir Gawain?

The Medieval tale of Sir Gawain and the Green Knight shows the importance of chivalry and honor.

The Green Knight, a pawn of Morgan Le Fay, challenges the Green Knight so as to play out the game which Le Fay has created so as to scare Guinevere.

Sir Gawain is a member of Arthur's Court. As a member of his court, he is required to uphold the code of the knight. When the Green Knight comes to Arthur's, during the Christmas season, he offers a challenge. Arthur, not willing to eat until entertained, allows the Green Knight to put forth his challenge. None of the knights step forward at first. Arthur, somewhat ashamed of his knights, initially accepts the Green Knight's challenge.

Gawain, knowing the he may fail and seeing himself as the weakest on one who will be least missed, decides to take Arthur's place. It is out of courtly love and chivalric duty by which he does this.

Le Fay's challenge, again as enacted by the Green Knight, is a simple ruse because of her hatred and jealousy against Queen Guinevere. Her whole plan is to frighten Guinevere so as to gain an upper hand over the Queen.