Once a week pulse out of small circuit for slow clock?
For a high precision timing function, consider using a real time clock with an integrated Temperature Compensated Crystal Oscillator (TCXO). An example is the Maxim DS3231 RTC with TCXO, available from eBay for under $3 a pair including international shipping.

This gives you some advantages over the crystal oscillators used on Arduino boards:


Accuracy of 3.5 parts per million, i.e. 110.45 seconds of drift per year.
Simplicity of counting time - you can set up the part to output a 1024 Hz square wave, for 0.977 millisecond precision for your clock ticks - or a once-per-second interrupt if you do not need the second decimal precision of ticking mentioned in the question
The inbuilt calendar allows a recalibration of start time every year, including leap years


To interface with such an RTC, you can use a very minimal microcontroller with an inbuilt RC oscillator, such as the ATtiny family.

The device thus would need merely the battery, the minimal microcontroller, the RTC chip, a few resistors and some decoupling capacitors, not much else I can think of - making it both inexpensive, and really simple to build. 

On a home-etched PCB, the entire device could be built in under an inch square, for perhaps $6 or so in singles, will run from a single button cell battery for years, and deliver an accuracy far beyond what any Arduino can do out of the box. A TCXO by itself would cost as much or more in single units.  

If you chose to build a few dozen of these, cost could be dropped down to well under $3 apiece including professionally etched PCBs with pretty silkscreen and resist layer.

Code:


The microcontroller uses I2C to set up the RTC for either 1 Hz or 1024 Hz output
An interrupt enabled pin on the MCU is triggered at each pulse set above, and counts duration - the microcontroller is set to low power sleep mode the rest of the time.
On achieving a count of 525948.76 seconds, or 525949 seconds if you are not fussy, the microcontroller pulses your external quartz clock.
Once a a month, or annually, the microcontroller retrieves the actual date and time from the RTC, and corrects for any drift due to the 525949 second rounding off