In the sentence "I gave my cricket the name of Bella", "of Bella" is the adjective phrase because it closely describes the word "name".
In the sentence "Searches on the Internet can turn up some unexpected results", the adjective phrase is "on the Internet" because they describe what kind of "searches" are in question, i.e. describe the type of searches.
Please, don't confuse possessive pronouns with adjective phrases (my cricket).