I think the preference for "the right" over "a right" in the Miranda warning reflects the tendency in the U.S. Constitution—and especially in the Bill of Rights—to prefer the former wording. The phrase "the right" appears five times in the ten amendments of the Bill of Rights:
Amendment I ... the right of the people peaceably to assemble, ...
Amendment II ... the right of the people to keep and bear Arms, ...
Amendment IV The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, ...
Amendment VI ... the right to a speedy and public trial, ...
Amendment VII ... the right of trial by jury ...
The wording "the right" also appears in amendments XII, XIV, XV, XIX, XX (twice), XXIV, and XXVI. The wording "a right" doesn't appear in any of the amendments.
The construction "a right against self-incrimination" would not be an illogical way to frame the guarantee that Amendment V expresses in these terms:
No person ... shall be compelled in any criminal case to be a witness against himself, ...
But we clearly match the voice of the other amendments more consistently when we express this guarantee as establishing "the right against [compulsory] self-incrimination."
Update (January 7, 2018)
A recent comment by site participant Chiwda ("I seriously doubt that the Miranda warning was constructed to match 'a tendency' in the US Constitution.") indicates that I may have done an inadequate job of explaining the relevance of the U.S. Constitution (and especially the Bill of Rights) to the wording of the Miranda warning.
First, let's look at the Miranda warning in its entirety, as given at MirandaWarning.org:
You have the right to remain silent. Anything you say can and will be used against you in a court of law. You have the right to an attorney. If you cannot afford an attorney, one will be provided for you. Do you understand the rights I have just read to you? With these rights in mind, do you wish to speak to me?
The warning explicitly identifies two rights as such: "the right to remain silent" and "the right to an attorney." And it implicitly identifies a third: the right of a criminal defendant to have court-appointed counsel if the defendant is too poor to be able to hire one of his or her own.
The Supreme Court case holding that police authorities were constitutionally required to apprise a person in police custody of those rights was Miranda v. Arizona (1966)—a case in which police had taken an indigent suspect into custody, interrogated him without first explicitly warning him of his rights to remain silent and to have access to an attorney (which the state would provide, if needed), and secured a confession from him. The Supreme Court reversed his resulting conviction.
The Miranda decision and the dissenting opinions attached to it use the expressions "a right" and "the right" quite often: "a right" appears 19 times in connection with individually specified rights under the constitution; "the right" appears 40 times in similar situations, and "the rights" appears 8 times in reference to multiple individually enumerated rights. So in their general discussions of the relevant law, the justices seem to have used the two forms interchangeably—at least to some extent.
With regard to the poster's question here, one of the most interesting aspects of the Supreme Court's majority opinion appears at 483–484 of the opinion:
A letter received from the Solicitor General in response to a question from the Bench makes it clear that the present pattern of warnings and respect for the rights of the individual followed as a practice by the FBI is consistent with the procedure which we delineate today. It states:
"At the oral argument of the above cause, Mr. Justice Fortas asked whether I could provide certain information as to the practices followed by the Federal Bureau of Investigation. I have directed these questions to the attention of the Director of the Federal Bureau of Investigation, and am submitting herewith a statement of the questions and of the answers which we have received."
"(1) When an individual is interviewed by agents of the Bureau, what warning is given to him?"
"The standard warning long given by Special Agents of the FBI to both suspects and persons under arrest is that the person has a right to say nothing and a right to counsel, and that any statement he does make may be used against him in court. Examples of this warning are to be found in the Westover case at 342 F.2d 684 (1965), and Jackson v. U.S., 337 F.2d 136 (1964), cert. den., 380 U.S. 935."
In short, the FBI's wording of its standard warning to persons whom it was holding for questioning mentioned "a right" to silence and "a right" to professional legal counsel. And yet, after Miranda, when the lower courts had to hammer out a constitutionally acceptable wording for an FBI-like warning to be given to all criminal suspects upon arrest, the warning that emerged and became known as the Miranda warning cited "the right" to remain silent and "the right" to an attorney.
Why the shift from "a right" to "the right"? As I said originally, I think that judges were influenced by the fact that the the parts of the Constitution that address individual rights against the federal government repeatedly use "the right to [or of] X" rather than "a right to [or of] X."
The constitutional rights that the majority opinion in Miranda v. Arizona was attempting to vindicate appear in the fifth and sixth amendments, which read in full as follows:
[Amendment 5 - Rights of Persons] No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.
[Amendment 6 - Rights of Accused in Criminal Prosecutions] In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the State and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor, and to have the Assistance of Counsel for his defence.
The fifth amendment right to silence is framed not as an affirmative right of the individual at all, but as a restraint on the power of the state to compel accused persons to incriminate themselves: "nor shall any person ... be compelled in any criminal case to be a witness against himself; ..."
The sixth amendment, however, explicitly presents the right to an attorney as an affirmative right: "the accused shall enjoy the right ... to have the Assistance of Counsel in his defence.
Significantly, whereas the FBI warning spoke of "a right to say nothing and a right to counsel," the Supreme Court in Miranda framed the requirements of a suitable warning as follows:
At the outset, if a person in custody is to be subjected to interrogation, he must first be informed in clear and unequivocal terms that he has the right to remain silent. ... The warning of the right to remain silent must be accompanied by the explanation that anything said can and will be used against the individual in court. ... Accordingly, we hold that an individual held for interrogation must be clearly informed that he has the right to consult with a lawyer and to have the lawyer with him during interrogation under the system for protecting the privilege we delineate today.
It is noteworthy that this portion of the Miranda decision appears more than ten pages before the opinion mentions (and quotes) the FBI warning. Evidently, even though the Supreme Court evidently approved of the FBI's warning, it had already shifted from "a right" to "the right" in its own formulation of what the warning should cover.
Given the original wording of the relevant constitutional protections and the repeated use of "the right to" or "the right of" in the Bill of Rights as a standard form for enumerating various other specific rights of the individual against the state, I think it was natural for courts to use the same wording to frame affirmative rights that the police must tell suspects that they possess, prior to attempting to interrogate them.
It is interesting that the right to silence appears in the Miranda warning affirmatively as "the right to remain silent" rather than negatively as "you cannot be compelled to testify as a witness against yourself"; but the FBI warnings (which the Supreme Court cited with evident approval) already framed the right as "a right to say nothing"—and once the choice of wording came down to "a right to" versus "the right to," it seems quite reasonable and consistent of the courts to have settled on the latter.