In Justice League: The Flashpoint Paradox movie, Thomas Wayne (Batman in this timeline) hands Barry a letter addressed to his son and tells him to go back in time and prevent this from ever happening. Later, Barry "wakes up" at his desk, much like the beginning of the movie, and gives Batman the letter from his father. How did he keep it while traveling through time?
More than likely, since the letter is simply a piece of paper, he put it under his costume while he traveled through time. The letter would be inside of his speed-force generated aura and thus protected from the ravages of travel the same way his costume is.
"The Flash's body is surrounded by what he calls his "speed force aura". This aura protects him and anyone who is running with him or carried by him, from the effects of using his speed. This includes friction and airborne particulate matter. It also protects him from injury from high speed impacts such as punches he delivers and receives from his opponents. With this aura, he is able to absorb kinetic energy. He possesses a level of superhuman resistance to injury which at times does extend far past normal physical interactions, as received from normal combat." Reference: Wikipedia - Flash's Powers
- The letter would survive the historical rewrite because the Flash is at the center of the event. He would have existed in both periods and experienced both timelines. So he would be able to carry away something small like a letter from one timeline to the next.
Actually, he wouldn't have to tuck the letter in his suit. He can travel through time with anything he can carry so long it's not heavy enough to slow him down. This is why he can pick people up and carry them out of danger without instantly ripping them apart. Ever hit something while driving? 30 mph to a dead stop? It hurts, a lot.