Home experiment to estimate Avogadro's number? How to get an approximation of Avogadro or Boltzmann constant through experimental means accessible by an hobbyist ?
 A: Your best bet is to try to replicate the experiments of Perrin who first measured Avogadro's constant.  This is a common lab in "Advanced Lab" courses in undergraduate or graduate courses, so you can probably find writeups and such via google.
The principle is to observe Brownian motion under a microscope and measure the diffusion constant.
Einstein's theory for Brownian motion relates the diffusion constant of a spherical particle to the temperature via the Einstein-Stokes law: 
$D=\frac{k_BT}{6\pi\eta r}$
Here $D$ is the diffusion constant, $T$ is the temperature, $\eta$ is the viscosity, and $r$ is the radius of a small spherical particle.  All of these properties should be measurable at least crudely with home equipment.
The way I did this in my lab course was as follows.  We had access to a microscope with a CCD camera which allowed digital video recording, as well as samples of monodisperse polystyrene particles (which are commercially available, and labeled with their size).
Suspend the particles in water (the viscosity of water is well known) at room temperature and place on a slide (better, put a thermocouple in your sample).
Take video of the particles Brownian motion, and then using something like John Crocker, David Grier, and Eric Weeks's celebrated particle tracking code extract 2D (or maybe 3D?) particle trajectories (i.e. $x(t)$, $y(t)$.  
Now plot the mean squared displacement of particles versus time.  The slope of this curve is the diffusion constant, which then yields an estimate for $k_B$ via Stokes-Einstein.
To recover Avogadro's constant, you need the ideal gas constant $R$, which is measured through independent means; typically via macroscopic thermodynamic experiments which probe the slope in $pV=nRT$.
A: Electrolysis.  Run a current through a weak acid, and measure the current going in and coming out.  Hydrogen ions in the acid will capture electrons and bond to each other to form hydrogen gas.  If you accept a measured value for the charge of an electron, you can find the number of hydrogen molecules liberated.  Then measure the volume of hydrogen formed at STP.  Avogadro's number is the number of molecules in a gas at STP with a volume of 2.2*10^-2 m^3.
A: Well, addressing only $N_A$, and allowing that it only gets you halfway there: Carl Sagan did a bit on Cosmos for estimating the size of a oil molecule. Put a known amount (both volume and mass) of cooking oil on a calm body of water and wait for the slick to diffuse to it's maximum contiguous area. Estimate the area.{*} Divide the original volume by that area and you have a rough value for the linear size scale of the molecule and thus how much volume it occupies. (We're neglecting issues of aspect ratio and polarity here so this is only a order of magnitude value.)
Now, if you have the molecular mass (from a mass spectrometer or some such) you're got all you need.

{*} A challenging step, but I'd use a photograph with some know length reference for scale and a planimeter. If you're not familiar with these instruments get ready to geek out. Very cool.
A: To get a rough estimate of the Avogadro number, one can also use a method similar to that used by Loschmidt ([1] http://iweb.tntech.edu/tfurtsch/Loschmidt/LOSCHMID.HTML). Gas viscosity can be measured (see, e.g., http://www.phywe.com/index.php/fuseaction/download/lrn_file/versuchsanleitungen/P3010201/e/LEC01_02_LV.pdf - gas flow through a capillary is measured there). Gas viscosity is equal (up to a coefficient) to a product of gas density, average molecular speed, and the mean free path [1]. As the average molecular speed is of the same order of magnitude as the sound velocity in gas, one can estimate the mean free path (I assume that it is not too difficult to measure sound speed). The size of a molecule equals (up to a coefficient) the mean free path times (liquid volume / gas volume) (http://en.wikipedia.org/wiki/Loschmidt_constant ).
EDIT (9/28/2013): It is not easy to measure volume of liquid air at home, as its temperature is very low, but the above measurement can be performed with CO2, as solid CO2 (dry ice) is easily available (it costs about a dollar per pound, as far as I know). It does not matter for our purpose that dry ice is solid, rather than liquid.   
