The original inspiration for this is Tinkernut's project. (Gerd Lobmeyer, Open Green Energy and others have done similar projects.)

• Power
• Size

I'm thinking of pulls from old mobile phones, esp. Nokia. Power use should be minimal. Availability (used) should be good.

• Nokia 5110 at SparkFun, 84×48, $10
• Nokia 5110 at Adafruit, 84×48, $10

What's available? Again power consumption should be small when backlight isn't used. Looks like finding something suitably sized with workable I/O connections is going to be difficult.

• 128x64 graphic
• 8x2 character
• eBay possibility
• ditto

What's the power consumption?

• White SPI I2C 0.96" w/breakout

• Adafruit Feather 32u4 Bluefruit LE, 2.0“ x 0.9” x 0.28“, 5.7 grams, battery charger, $30. Possibly all FOSS.
• Adafruit Feather M0 Bluefruit LE, 2.0” x 0.9“ x 0.28”, 5.7 grams, battery charger, $30. Possibly all FOSS.
• Bledino, 1.7“ x 0.8”
• RFDuino, 0.6“ x 0.6”, $17.99, $14.99

• DigiSpark Pro, 1.1“ x 0.72”, $10.
• Femtoduino, $35
• Microduino
• Pro Micro, 1.3×0.7“, $19.95 (official)
• Pro Mini 0.7×1.3”, $3 (clone) to $10 (official)
• Qduino Mini, battery charger, $30.
• Tinyduino, battery support, goofy I/O $20

• Use a Nordic BLE module for everything?

• HC-06: $TODO, TODO mA, size is a bit big.
• HC-05: $TODO, TODO mA, size is a bit big.

• Adafruit Bluefruit LE SPI Friend, 0.9“ x 1”, possibly FOSS, $17.50
• Adafruit Bluefruit LE UART Friend, 0.8“ x 1.26”“, possibly FOSS, $17.50
• Itead HM-11, 0.7” x 0.53, $8.50
• Itead HM-10, 1.1“ x 0.5”, $6.50
• ReadBearBLEmini
• Seeed Bluetooth V4.0 HM-11 BLE Module, 0.5“ x 0.7”, $13.
• Seeed Micro BLE Module, $17.

• Lipo Charger Module for Arduino
• Details about DD08RCRMA Mini 5V 1A Lithium LiPo Charger Module for Arduino