Hardy gout tubers are obtained during the My Arm's Big Adventure quest from Murcaily. They can not be dug up or planted in a herb patch. One is given to the player after he or she gets 60% favour, but then Murcaily gives the player a handful after My Arm goes on a rampage.
They are given to My Arm towards the end of the quest, before the final fight with the Baby Roc and Giant Roc. After the quest, if you try to pick the Hardy goutweed from My Arm's patch, then you'll get the message, "That's My Arm's problem now."