This was a very good question that was raised. In attempting to research an answer, I came across some very interesting data points.
-- The FBI falls under the umbrella of the Department of Justice, and it is the DOJ's responsibility to investigate and prosecute cyber-crimes. The FBI is the agency which the DOJ primarily uses to achieve these ends. See the excerpt below, taken from: http://newsroom-magazine.com/tag/fbi-cyber-division/
*** 'Prosecuting cybercriminals and identity thieves: One of the Department of Justice’s core missions is protecting the privacy of Americans and prosecuting criminals who violate that privacy.' ***
-- Since 9/11, the FBI has put much more emphasis on investigating and prosecuting cyber-crime in addition to other duties.
-- Sony reached out to the FBI when they learned of the breach, probably because they understood its role in such an investigation, and possibly because of the overall involvement of Anonymous in the DDoS portion of the attacks.
-- There are a number of different government agencies looking into this Sony matter, beyond just the DoJ and FBI. Quite a bit of European data was accessed, which might put Sony on the wrong side of a number of government groups with strong(er) privacy laws.
See the following for some more details: http://www.gamespot.com/news/6310487/sony-enlists-fbi-as-us-canadian-authorities-look-into-psn-breach http://www.vg247.com/2011/04/29/homeland-security-lends-a-hand-in-psn-data-breach-investigation-sony-changes-faq/ http://www.dispatch.com/live/content/business/stories/2011/05/09/fbi-probing-consumer-data-breach-at-sony.html?sid=101
Wikipedia also has a very nice article on the mandate of the FBI, which provides insight into its role. As the breach was executed against US-based information technology assets, and involves the possible theft of data belong to US citizens/residents, it falls nicely within the FBI's jurisdiction.