Get your 5V out of a single 3.7V LiPoly / LiIon cells at a MINIMUM of 500mA. Recharge while drawing power on-the-fly.
Sure, you can get regular "powerbank" modules to recharge your dead electronics, but Adafruit has engineered this really well thought-out rechargeable 5.2V power module. This is like their PowerBoost Basic, with LiPoly charging circuitry added in.
Plug your LiPoly cell into this, then power up your load (like your Raspberry Pi). When your battery power starts to drop, plug in the microUSB charger to the module and recharge your battery at 500mA all while keeping power to your project! It's a mini-uninterruptable power supply (assuming your load pulls less than ~300mA).
First of all, why 5.2V instead of a solid 5V? The extra 0.2V will adequately compensate for any connector and cable losses between this module and the project you are powering. Even so, feeding 5.2V into a "5V" designed device is quite safe. Feel comfortable using this to power your Arduino / RaspPi / BeagleBone / etc...