If you're happy to use an ATMEL AVR device, then you'd need:
* ATMEGA48/88/168/328 (just one of these) Revision D or later (or the 32kHz TOSC is no good),
* a 32kHz crystal for the RTC,
* an HD44780 LCD (I'm assuming you're OK with one of the 2x16 character modules..???),
* a switch,
* some capacitors e.g. 2x 16pF, 1x 10uF, 2x 100nF,
* a programmer, e.g. USBISP or USBASP
* batteries and holder, and
* a programming connector (e.g. 2x5 or 2x6 DIL pin header).
This can be programmed easily using AVR-GCC or WinAVR C compilers.
Regarding saving the numbers and timestamps, the program memory can be used for this purpose; just note that blocks of 32 bytes should be written at a time (RAM or EEPROM can be used for short-term storage of this data).