Cost and Power of EEPROM and Flash?
FeRAM (F-RAM, FRAM) is probably the lowest power writable non-volatile storage around. It can be rewritten without having to worry about limited numbers of writes and it is very fast (read and write.) And with only a need of 1 kB total, it's the way I'd lean for coin cell use. You can either select from Cypress or TI for inexpensive parts that are generally available. TI includes MSP430-based microcontrollers which include enough for your use, too. You'll spend about $1 for a Cypress part (memory only) and you'll spend about about the same to get the memory with a TI MSP430 microcontroller added in, too.

(There is also MRAM, but it is very expensive and only available in small quantities and hard to find, besides. I'd avoid it for now.)