small inexpensive battery powerable component for daily timer circuit with LED and button
The usual approach here is 


Use a microcontroller with integrated RTC
Use a lithium coin cell, e.g. the CR2032, they have an incredible storage life, and you really don't need much power
make sure to pulse the LED, which gives much perceived brightness per energy used


To that end, something like an ATMEL SAMD10 should have about 10 µA of standby current while still keeping the RTC on, and can run directly off the battery.

How to program that: good question; your 15min off per week is a pretty relaxed constrain (because: 15min/(60min/hr·24hr/d·7d)= 6.67% tolerance), so that reprogramming could be pretty rare. If you can just press a button that means "remind me exactly now, every 24hrs", that would probably be easiest to code :)