What you start with is not too critical as you can develop basic concepts on any Arduino hardware and use it to drive a display of your choice for testing. This can be migrated to whatever is found suitable as the project progresses.
Important to consider are the number of I/O lines of various sorts that you need. Most AVR ICs that you are liable top use will have "some" analog input and PWM output lines. If you were going to drive a "bare" LCD you may need to use a processor with LCD drivers, but in most designs use of an existing module with eg SPI interface is probable preferable.
Most Arduinos in the UNO, Nano, Pro Mini, ... range are in reality not a lot more functionally that a processor IC, USB bridge (if needed) and voltage regulator(s). For a tidily small watch you may ultimately wish to use a bare SMD AVR IC with an Arduino boot loader code installed. You can move away from AVR ICs to other Arduino compatible systems - with some extra knowledge needed in most cases.
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Examples only:
Here is an Internet of things hardware roundup which includes some small controllers as examples.
Panstamps offer MSP430 and ATMega328p (Arduino compatible versions).
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RFDuino - from here offer some fingertip sized products with RF capability. Note that this use an ARM Cortex M0 processor, not an ATMega controller. While it is "Arduino compatible" itwillit will not run all code that the typical UNO, Nano, ... boards run.
Manual here
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Their kickstarter page https://www.kickstarter.com/projects/1608192864/rfduino-iphone-bluetooth-40-arduino-compatible-boardTheir Kickstarter page
