Tynes, a free agent, helped the Giants win two Super Bowls by nailing the game-winning overtime kicks in the 2007 and 2011 NFC Championship Games. He's the only kicker in NFL history with two overtime game-winning playoff field goals.
Brown, 33, played in four games last year for the Cincinnati Bengals, converting 11 of 12 field goals and all eight of his extra point attempts.
"Josh is an experienced, proven kicker in this league," general manager Jerry Reese said in a statement released by the team. "Just like we try to do at every position, we expect great competition for the placekicking job, and that will only make us better."
Over his 10 seasons, Brown has made 231 of 284 field goal attempts (81.3 percent). He spent his first five seasons with the Seattle Seahawks before playing for the Rams from 2008 to '11.
"We bring a lot of opportunities to score from a great distance," Brown said on Giants.com of what he can do with his new team. "We've kicked a number of 50-plus yarders and feel very comfortable from 55 and in, even in harsh winters and that and a lot of confidence."
Tynes made 33 of 39 field goals last season -- his sixth season with the Giants. He has converted 81.5 percent (190-of-233) of the field goals he has attempted in his nine-year career.
With salary cap considerations, the Giants have been making minor moves in free agency thus far. They lost tight end Martellus Bennett to Chicago on Tuesday but added cornerback Aaron Ross. The Giants' biggest move so far came before free agency began when they re-signed left tackle Will Beatty and added former Philadelphia Eagles defensive tackle Cullen Jenkins.
The team's priority in free agency is to re-sign free-agent left guard Kevin Boothe. The Giants are also continuing their efforts to improve defensively as they are hosting Minnesota linebacker Jasper Brinkley on Wednesday, a source confirmed.