How to Measure Power Consumption on Extremely Low Power Devices?
One solution is to use an instrumentation amplifier to measure the voltage drop across a shunt resistor. These are designed to offer an extremely high input impedance to both inputs of the amplifier (in excess of 1 giga-ohm), while allowing you to amplify this signal by relatively large factors (1000x is not uncommon). Note that the fact that there is a really high input impedance isn't too terribly important for this particular application, however the high amplification factor is.

The basic schematic looks like this (I'm using IA is a self-contained package for an instrumentation amplifier; often, these have an external gain resistor so you can choose whatever gain you want):







simulate this circuit &ndash; Schematic created using CircuitLab

The large amplification factor allows you to use a relatively small sense resistor, mitigating a large portion of the effect of the burden voltage on your DUT.

If you're just looking to buy an off-the-shelf solution which does effectively this, you could look into something like the uCurrent. There are probably also specific IC's designed for this current range.

Since the outputs of these type of current sensors is just a relatively isolated analog voltage, you can use any standard oscilloscope or voltage meter to measure the current.

These very simple devices are good enough for things in the nano and micro ampere ranges and are relatively easy to use.

For even smaller currents (pico or fempto ampere ranges), there are specially designed chips such as the LMP7721, along with a few pages of application notes on low current design. It's unlikely you'll want something like this for measuring power current draw. These are typically used by the scientific community for measuring sensor outputs (photodiodes/other very low current sensors).