It has been 15 years since Marc Andreessen developed the Netscape Internet browser that introduced millions of people to the Internet. After its early success, Netscape was roundly defeated by Microsoft in the so-called browser wars of the 1990s that dominated the Web's first chapter. Mr. Andreessen appears to want a rematch. Now a prominent Silicon Valley financier, Mr. Andreessen is backing a start-up called RockMelt, staffed with some of his close associates, that is building a new Internet browser, according to people with knowledge of his investment.