Adding a little to the good information in the other answers...

I think key to seeing why the answer is "1" and not "2" (Steve warns Jim, lightly, not to mind) is that there is an elided "I" at the beginning of the idiom. That "I" doesn't have to be elided. See e.g. the song 

    I entered the kitchen, 'twas cosy and bright
    Soon a fine hearty supper, I put out of sight
    Says she "Have a drop of the old mountain dew"
    And me darlin', says I, I don't mind if I do

(http://www.shamrocksinthewind.com/lyrics_frolickn.html)
The Rovers perform this song at http://www.youtube.com/watch?v=EKi00SVmLJ4

The other aspect to understanding the form of this idiom is typical British [humorous understatement][1]. So the rather weak "I don't mind" is understood as humorous understatement for "Gladly!"


  [1]: http://en.wikipedia.org/wiki/Understatement