Firstly we need to know whether you want a
highly precise, or a highly accurate current source (or some mix of both).
A highly precise 10uA current source might give you 9.5410 uA, but vary only by 0.001 uA over some load and line variation, temperature and time.
A highly accurate 10uA current source may give you 10.001uA under prescribed conditions of temperature, load, etc., but vary by 0.5uA under other circumstances.
In general, accuracy is improved by allowing an adjustment of the output current (in this case), whereas precision is enhanced by using highly stable components. They are somewhat at odds because the device used to trim the current may not be highly stable and thus may decrease precision.
OK, lets assume you want something realistic, perhaps within half of a percent or better.
To do this you need to ensure that the sum of all your tolerances (well, the ones which matter) are less than 0.5% in both absolute value, and over the temperature range you're contemplating (and also over time)
The simplest circuit uses an op-amp (driving perhaps a mosfet or bipolar transistor for higher currents) to maintain a fixed voltage across a sense resistor. The voltage it is compared against comes from a voltage reference.
The sources of loss of accuracy and precision are:
1) voltage reference accuracy and precision
2) voltage divider (if used) for voltage reference
3) input offset of op-amp
4) sense resistor accuracy and precision
5) op-amp input current (primarily from sense voltage)
Add to this the accuracy and precision of any current measuring device you have if you are going to calibrate the current source.
Let's say we can use a 2.5V 0.3% voltage reference. And lets say we can pick a 0.1% resistor that will drop 2.5V at the required current -- we've used up 0.4% already. We need to ensure that the effects of the input current and offset voltage of the op-amp do not add more than 0.1% more inaccuracy. Basically that means the input current for the op-amp needs to be as low as possible (0.05% of the set current) and the offset voltage less than 0.05% of the sense voltage.
Of course, if you know the characteristics of the op-amp, you may be able to allow for a relatively constant input current, and you may be able to trim away some of the offset voltage, but you still have to consider their variances over temperature (at least) and if you are trimming things, the accuracy and precision of your trimming.
So yes, the question remains: what do you mean by precision?