For an idea, we pick the Planck units (with other constants thrown in). For those who don't know, the Planck units [footnote 1] are:

A system of units needs a set of fundamental units, from which all the other units can be derived. In SI, these fundamental units are:

Now for our fundamental length, mass, and time units we can just use the Planck units directly. We will call these units L, M, and T respectively.

For electric current, we can simply use charge divided by the Planck time, but we need a charge. An obvious (unit independent) choice is the magnitude of the charge on an electron, e.

For thermodynamic temperature, we need to relate a fundamental constant concerning temperature to the rest of the units. An obvious choice is the Boltzmann constant, which has units of energy divided by temperature. So we can choose our energy unit (which we will discover later) divided by the Boltzmann constant (k) as our temperature unit.

Luminous intensity is a tricky one. There are only three luminosity units in SI (cd, lm, lx), only one of which is fundamental (cd). Since these are specialized units that are only intended to be used under extremely narrow situations (this is clear when you look at the definition of the candela), we can make the sweeping generalization that all applications using these units can in fact use the derived units for energy and power (combined with other units). For instance, the Planck equivalent of the candela would just be the power unit; the lumen equivalent would be the power unit times the steradian, and the lux equivalent would be the power unit times the steradian divided by the square of the Planck length.

Then we have amount of substance. This one's easy; we eliminate the unit. Wherever mol would have appeared, it is just replaced with the appropriate number (6.02 x 10²³).

So now we have our fundamental units (less the luminosity units), with SI conversions:

Now we can start getting derived units. There are an awful lot of derived units, but only some of them are named. (For instance, area, volume, density, momentum, entropy, viscosity, intensity, etc., are all perfectly good -- and even frequently-encountered -- units, but they happen to have no unique names in SI.) For simplicity's sake, we'll only go over the derived units which have names in SI (with conversion factors). First, the mechanical units:

Note what we've really done here: We've made a system of units out of the fundamental constants hbar, G, c, k, and e, with some choices of units, but nothing so arbitrary as picking random values for units.