Answer:
The word in the sentence that the prepositional phrase "on the counter" relates to is "placed".
Step-by-step explanation:
To place is a ditransitive verb. Ditransitive verbs take both a direct object and an indirect object. In the sentence above, the verb selects the NP "her invention" and the PP "on the counter". The NP denotes the thing located, and the PP the location of the thing. We can prove that place is a ditransitive verb that requieres these two arguments (apart from the subject) by taking out one of them, as this would create an ungrammatical sentence (see 1).
1) *She carefully placed her invention