The pinoccio dev board consumes around 17mA when running at full blast. ( From the faq page of pinocc.io )

The Nokia 5110 LCD module would take around 40mA with the back light LED on.

Assuming I use an 8 Ohm ,  0.5W speaker. I would consume around 250mA

Adding a buffer of 20mA for other components.

Totally we have around 330mA.

So, all the above would be easily powered using the 550mAh battery supplied with the pinocc.io dev board for atleast an hour.

Given that the speaker would be used intermittently and the back light LED of the LCD module would be used only at night. We should still be able to get around 4 or 5 hours of usage from this device on a single charge.