The Weed badge is awarded to players who have pulled a vast amount of weeds growing in their town, or in the towns of other players. Weeds pulled in dream towns via the Dream Suite will not contribute to the total required for each badge.
While I can't help you (I pull em everyday too) I can give advice. Pulling wilted flowers counts too. I recommend getting a mess of flowers from the island minigames, and not watering them. They won't all wilt overnight, but it will give you an extra 10 or so weeds a day. Provided it hasn't rained/snowed the previous day.