A lot of people seem to think so, but on balance I find it less than likely. While Brown has a reputation as an "extreme conservative," it's more accurate to say that she is a hard core libertarian. A hard core libertarian is not likely to vote in a consistently conservative way, which I think makes it less likely that Bush will nominate her to the Supreme Court.

  For example, I think Brown would be more liberal than any sitting Supreme Court justice on questions of criminal law and procedure, a subject that may carry over to the war on terror and the scope of executive power. Criminal law cases occupy maybe 25% of the Supreme Court's docket, and my quick review of a few Brown opinions suggests that Brown would side on the "liberal" side (generally speaking, more pro-defendant than the norm) in such cases. The handful of Brown cases I read in the area of criminal law have a William O. Douglas flavor to them; they're passionate about the meaning and importance of the Bill of Rights, and very critical of what she sees as the modern Supreme Court's watering down of its protections.

  An example is Brown's separate opinion in People v. McKay, 27 Cal.4th 601, 117 Cal.Rptr.2d 236 (2002), in which she offered the following take on the Fourth Amendment in a case about an arrest of a biker for cycling the wrong way down a one-way street:

  . . . The framers [who enacted the Fourth Amendment] sought to preclude "the petty tyranny of unregulated rummagers." (Amsterdam, Perspectives on the Fourth Amendment (1974) 58 Minn. L.R. 349, 411. The first clause of the Fourth Amendment issues a global command: "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated...." The second clause--specifically prohibiting the issuance of a warrant except "upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized"--addresses the narrower compass of traditional search warrants "for contraband, stolen goods, and the like." (Amar, Fourth Amendment First Principles (1994) 107 Harv. L.R. 757, 765, fn. omitted.) Its purpose was probably not to make warrants mandatory, but to limit the opportunity of the executive to obtain warrants in the first place. (See Taylor, Two Studies in Constitutional Interpretation (1969) pp. 38-50.)
  Unfortunately, the Supreme Court's modern Fourth Amendment jurisprudence gives new vigor to petty rummagers. In analyzing searches incident to arrest, the court has tended to equate probable cause with reasonableness, but these terms "serve distinct functions, which are lost by homogenization of the legal vocabulary." (Gramenos v. Jewel Companies, Inc. (7th Cir.1986) 797 F.2d 432, 442.) In Whren, supra, 517 U.S. 806, 116 S.Ct. 1769, 135 L.Ed.2d 89, Justice Scalia authored a unanimous opinion in which the Supreme Court ruled that when a police officer observes a traffic violation, stopping the vehicle is reasonable and the officer's subjective motivation plays no part in the Fourth Amendment analysis. (Id. at pp. 810, 813, 116 S.Ct. 1769.) Whren essentially legitimized pretextual stops--the sine qua non of unjustified and arbitrary law enforcement. A pretext stop occurs when "the justification proffered by the State for an arrest is legally sufficient, but where the *632 arresting officer was in fact making the arrest to search the arrestee incident to arrest for a reason which was legally insufficient to support the arrest." (Burkoff, The Pretext Search Doctrine: Now You See It, Now You Don't (1984) 17 U. Mich. J.L. Reform 523, 523.)
  The court has clearly seen the need to curb police discretion only when law enforcement agents search and seize without probable cause (Prouse, supra, 440 U.S. at p. 663, 99 S.Ct. 1391), and when police operate under vague enforcement standards which confer a virtually unrestrained power to arrest. (Kolender v. Lawson (1983) 461 U.S. 352, 360, 103 S.Ct. 1855, 75 L.Ed.2d 903 (Kolender ).) In Prouse, an officer randomly stopped the defendant to check his driver's license and registration. The United States Supreme Court held that subjecting drivers to random checks, without reasonable suspicion, is unreasonable under the Fourth Amendment. (Prouse, at p. 663, 99 S.Ct. 1391.) The Prouse court could not "conceive of any legitimate basis upon which a patrolman could decide that stopping a particular driver for a spot check would be more productive than stopping any other driver. [Moreover,] [t]his kind of standardless and unconstrained discretion is the evil the Court has discerned when in previous cases it has insisted that the discretion of the official in the field be circumscribed, at least to some extent." (Id. at p. 661, 99 S.Ct. 1391.) Whren distinguished Prouse on the ground that the spot-checking officer did not have " 'probable cause to believe that a driver [was] violating any one of the multitude of applicable traffic and equipment regulations' " (Whren, supra, 517 U.S. at p. 817, 116 S.Ct. 1769), whereas in Whren, the arresting officer did have probable cause to believe that the defendant had violated a traffic law.
  . . . The United States Supreme Court purportedly believes that " 'observed [traffic] violations' ... afford the ' "quantum of individualized suspicion" ' necessary to ensure that police discretion is sufficiently constrained.'' (Whren, supra, 517 U.S. at pp. 817-818, 116 S.Ct. 1769.) In reality, an officer's discretion in deciding whom to stop is not constrained at all by a probable cause prerequisite because the officer need only point to a minor traffic violation to negate a claim of unfettered arbitrariness. (1 LaFave, Search and Seizure (3d ed. 1996) § 1.4(e), p. 123.) Due to the widespread violation of minor traffic laws, an officer's discretion is still as wide as the driving population is large. In the pervasively regulatory state, police are authorized to arrest for thousands of petty malum prohibitum "crimes"--many too trivial even to be honestly labeled infractions. They are nevertheless public offenses for which a violator may be arrested. Since this indiscriminate power to arrest brings with it a virtually limitless power to search, the result is the inevitable recrudescence of the general warrant. (Salken, The General Warrant of the Twentieth Century? A Fourth Amendment Solution to Unchecked Discretion to Arrest for Traffic Offenses (1997) 16 Pace L.Rev. 97, 146.)
  An officer's observation of a very minor offense authorizes him to stop the car (Whren, supra, 517 U.S. at p. 817, 116 S.Ct. 1769) or bicycle (United States v. McFadden (2d Cir.2001) 238 F.3d 198), arrest the driver or rider (Atwater, supra, 532 U.S. at p. 354, 121 S.Ct. 1536; McFadden, at p. 204), search the driver or rider ***262 (Robinson, supra, 414 U.S. at p. 235, 94 S.Ct. 467), search the entire passenger compartment of the car including any package inside (Belton, supra, 453 U.S. at p. 460, 101 S.Ct. 2860), impound the car and inventory all of its contents (Colorado v. Bertine (1987) 479 U.S. 367, 374, 107 S.Ct. 738, 93 L.Ed.2d 739), and imprison the offender for up to 48 hours (Atwater, at p. 352, 121 S.Ct. 1536; County of Riverside v. McLaughlin (1991) 500 U.S. 44, 56, 111 S.Ct. 1661, 114 L.Ed.2d 49).
  Thus, after Atwater, the notion that "[a]n individual operating or traveling in an automobile does not lose all reasonable expectation of privacy simply because the automobile and its use are subject to government regulation" (Prouse, supra, 440 U.S. at p. 662, 99 S.Ct. 1391) is simply no longer true. In fact, the same rules apparently apply to those who walk, bicycle, rollerblade, skateboard, or propel a scooter. Probable cause is ubiquitous.
  Given the pervasiveness of such minor offenses and the ease with which law enforcement agents may uncover them in the conduct of virtually everyone, the probable cause requirement is so diluted it ceases to matter, "for ... there exists 'a power that places the liberty of every man in the hands of every petty officer,' " precisely the kind of arbitrary authority which gave rise to the Fourth Amendment. (1 LaFave, Search and Seizure, supra, § 1.4(e), p. 123, quoting 2 Wroth & Zobel, Legal Papers of John Adams (1965) 141 142.)

  (emphasis mine)

  It's hard to know if Brown's libertarian wordview would carry over to war on terror questions, but they might. Given the perceived importance of such questions in the future, I think it's somewhat less than likely that Bush would nominate Janice Rogers Brown.

  UPDATE: Is People v. McKay unrepresentative of Brown's views in the area of crimal law? In the comments, Edward A. Hoffman argues that it is.