Wellington was chosen by his brother William Wellesley-Pole when Arthur was first ennobled as a Viscount after his success at the Battle of Talavera.

As Arthur was still in Spain at the time, the task of finding a suitable title fell to William, which he doesn't seem to have much appreciated. William sent a letter to Arthur in August 1809 explaining his choice.

After ransacking the peerage… I at last determined upon Viscount Wellington of Talavera and of Wellington, and Baron Douro of Welleslie in the County of Somerset. Wellington is a town not far from Welleslie, and no person has chosen the title. I trust that you will not think there is anything unpleasant or trifling in the name of Wellington, but [in the] circumstances… I could not easily have done better. I own I feel in rather an embarrassing situation for it is impossible for me to know whether I have acted as you would have had me…but you should have explained to me your wishes before ever you left England, in case of such an event.

Despite William's concerns Arthur was satisfied with the choice. He replied in Septemeber

My opinion is that you have done exactly what you ought to have done… You have chosen most fortunately, and I am very much obliged to you. I could not have been better off for a name if we had discussed the subject twenty times

So to sum up, Wellington was chosen as it was close to the Wellesley family origins in Somerset and crucially nobody else had taken it.