@AJB: I am sorry, you may get frustrated again, because both are grammatically correct. :)
In "As you go through various settings, you will have the option to allow certain activities...," to allow is an infinitive-adjective which describes the noun "option." As adjective, it answers the question "What [kind of] option?" Answer: (the option) to allow
For lessons on the use of infinitive as adjective, please see this: http://www.chompchomp.com/terms/infinitive.htm
On the other hand, in "...you will have the option of allowing...," of allowing is a prepositional phrase composed of the preposition "of" and the gerund "allowing." Like in the previous case, "of allowing" functions as an adjective which answers the question "What [kind of] option?" Answer: (the option) of allowing
For lessons on the use of gerund as object of preposition to form a prepositional phrase which in turn can function as adjective or adverb, please see this: http://examples.yourdictionary.com/prepositional-phrases-examples.html