For the "box", you can make a mold and use fiberglass, too. Thin weaves will work with small enclosures.
For the LCD: the ones in bike computers of old are specific made for the job, and obviously, they use less energy. The most energy efficient LCD is, e-ink. Newer ones have very high refresh rates compared to older generations. They are quite cheap, too. Instead of Arduino, I'd use something simpler like PIC12, maybe. ATMEL's sleep or low power modes are not that efficient, or cannot be controlled perfectly as in PIC's. PIC's are more efficient.
As a battery alternative, you can use rechargeable LıPo batteries - they' re cheap and comes in sizes that can fit inside a watch. If you hook up a TP4056 module, you can charge them under an hour!