New York Gov. David Paterson (D) will end the suspense Friday and name a successor to former Sen. Hillary Rodham Clinton (D), who became secretary of State this week.

Paterson has scheduled a midday news conference in Albany to announce his choice.

But on the heels of Caroline Kennedy’s abrupt and unexpected withdrawal from the Senate sweepstakes early Thursday, controversy continues to swirl around the replacement process.

On Thursday, Rep. Carolyn McCarthy (D), who rose to political prominence after the shooting death of her husband in 1993, told the New York Times that if Rep. Kirsten Gillibrand (D) is appointed to the Senate, she would mount a primary challenge next year against Gillibrand.

Gillibrand and state Attorney General Andrew Cuomo (D) are seen as the leading contenders for the Senate appointment now that Kennedy has departed from the scene, though several other officeholders are also under consideration.

McCarthy, who has made the push for stricter gun-control measures the central cause of her Congressional career, cited Gillibrand’s support from the National Rifle Association as the reason why she would run against her House colleague. McCarthy’s husband was one of several people killed on a Long Island Railroad car during an evening rush hour when a gunman opened fire on the crowd. Her son was injured in the shooting spree.

“To have a Senator representing the NRA for New York, that would be wrong,” McCarthy told the Times. “If it comes down to that, I will primary in 2010.”

McCarthy said she would alternatively support a younger candidate who supports gun-control measures if someone else came forward to run against Gillibrand.

McCarthy’s remarks come on the heels of a piece published in the liberal Village Voice newspaper Thursday questioning Gillibrand’s Democratic credentials. The article touched on Gillibrand’s voting record on gun-control issues but also described the close ties between Gillibrand’s father, Albany lobbyist Douglas Rutnik, and several prominent New York Republicans, including former Sen. Al D’Amato and former Gov. George Pataki.

It also suggested that Gillibrand’s first Congressional race against then-Rep. John Sweeney (R) was secretly aided by Pataki allies. Gillibrand represents a conservative upstate district and has been unabashed about her support for gun owners’ rights.