Final answer:
The component that guarantees individuals the right to their own religious beliefs within the First Amendment is the Free Exercise Clause, which operates alongside the Establishment Clause to protect religious freedom in the U.S.
Step-by-step explanation:
The subject of the question is the 1st Amendment guarantee of religious freedom which ensures individuals have the right to believe as they choose in matters of religion. To specifically answer the question: the component that guarantees to each person the right to believe whatever he or she chooses to believe in matters of religion in the First Amendment is the Free Exercise Clause.
This clause is directly contrasted with the Establishment Clause, which prohibits the government from establishing an official religion or showing preference for one religion over another. The First Amendment as a whole safeguards several key freedoms, including the freedom of speech, press, assembly, and the right to petition the government for a redress of grievances, alongside the two religious clauses.