It could be Lews Therin's sword from the War of Power which was unearthed directly, or found in a stasis box.

1. If it's Power wrought it could have survived 3,000 years. Centuries don't preclude it from being millenia old... it's merely 30 centuries old.

2. He's surprised that it's from his own memories because the theme of the chapter is that the two men are merging mentally. Later he shouts directly at Lews Therin, which is also a first. Instead of it being read as a memory of Rand's, it might be his surprise that Lews Therin said nothing and that he recognized it. In essence, they're becoming one man.

It can't plausibly be a False Dragon's. Amalasan is no more a valid memory to Rand than it is to Lews Therin. Rand is 21 years old and he doesn't have Mat's memories. He's got his life experiences and Lews Therin's. It must logically be Lews Therin's that he's referring to, but he's just surprised that Lews Therin isn't the one mentioning it.