Simple Experiment to Demonstrate Special Relativity I am trying to think of a good experiment that can be done for under $250 or so that would demonstrate some aspect of Special Relativity. Ideally this will be done in a few years with my kids when they were a bit older. 
 A: You must know that special relativity in the sense of increasing masses and time dilation happens with high energies, that can only be found in the high energy laboratories. CERN has teaching material on its site that will be useful for high school children.
The only simple "experiment"  I can think of is that of using old bubble chamber photographs of interesting 4 constraint events. For example this event of an antiproton annihilating into a lambda anti lambda pair.
One could have the child measure the curvatures to find the momenta, that would be the experimental part, identify  the type of particle from the ionisation density, etc.
This would demonstrate annihilation, the mass energy relations etc.
A: The historically accurate way is to do Fizeau's experiment. This requires setting up an interferometer, but I am sure that commercial interferometers, being precision instruments, are very expensive. To make an interferometer, you should by a laser and a mirror, and some optics splitters and make it yourself--- it doesn't have to move. You make the laser go through a glass pipe into water, and you slowly make the water move at a greater rate, and look for the interference fringes moving as the velocity of the water changes. From this you can infer the speed of light in a moving medium, and this is a good test of the relativistic velocity addition law. If you built it yourself (it's a massive project), the material cost might be less than $250.
Another possibility is to look for the  or measure the cyclotron frequency of an electron in a magnetic field. You make the magnetic field stronger, to see the change in mass with velocity. This was an early test that discriminated between SR and other contemporary theories.
In graduate labs, muon lifetime is a standard measurement, and then the muon arriving on the ground is a relativistic time-dilation effect. I am not sure about the cost. This one is only qualitative, since you aren't measuring the muon lifetime as a function of velocity. That would be too expensive.
Perhaps the best and cheapest way to motivate relativity is to demonstrate that a magnet moving past a loop of wire gives the same induction current as a wire moving past a magnet. The two effects have different explanations without relativity: the wire moving feels a Lorentz force from the magnetic field that drives the charge carriers, while the moving magnet must generate an electric field to cause a current, by Faraday's law of induction. Yet the two effects give the same magnitude of current, which suggests that E and B rotate into each other under a transformation of frame, something which only happens in relativity. This can be done for a few dollars.
A: Another idea is the demonstration of the principles behind GPS system. You can demonstrate that you can determine your position, exploiting just four satellites. GPS does it by solving the following system:
$$|\vec{r}-\vec{r}_i| = c(t-t_i)$$ 
You have four unknowns: your position $\vec{r}$ and your precise time $t$. So you need four equations to close your system -- four satellites with positions $\vec{r}_i$ and signal emmition times $t_i$. (I've actually seen somewhere that if you can get raw data from the GPS, then you'd be able to calculate your position "by hand" by solving this system.) 
Notice that one doesn't need to correct for relative speed of the receiver and the satellite. You just put the speed of light $c$ in the equations, despite the fact that this relative motion can be really fast. The reason for that is Special Relativity.
Here is the review on that topic in Physics Today.
Also check this video for nice simple introduction.
A: You could show them a piece of gold. Maybe drop it in acid and observe the (lack of) result.
Gold owes both its color and corrosion resistance to special relativity. See What Gives Gold that Mellow Glow?
A: Like anna v, I'm a particle physicist and am thinking along those lines.
I have not tested the following and it would require some electronics work from you. Further, this is not a classroom demo, but a lab project in which the students would be able to convince themselves that relativity works as advertised.
Examine the relationship between the $T$ and $p$ for an electron
Building the apparatus1


*

*Obtain a cathode ray tube with a luminous screen at the far end. The longer the flight path the better. People are all but giving away CRT TVs and monitors these days, but it would be better to get one for which detailed technical manual were available (an antique analog oscilloscope would be a good bet) as you will have to rework the drive electronics.

*Rip out the electronic driver and replace it with one where you can know and vary both the accelerating potential and the steering current (only one direction of steering is necessary). Both supplies will need to be stable and highly accurate.
The lab work


*

*By way of theory, demonstrate that $T \propto \Delta V$ and $\Delta \theta \propto \frac{I}{p}$ (where $T$ is the kinetic energy of a single electron, $V$ is the accelerating potential, $\Delta \theta$ is the angular deflection of the beam and $I$ is the current through the steering coil).

*The students should take detailed measurements of $\Delta \theta$ as a function of both $I$ and $\Delta V$. (I generally prefer to ask them to develop the methodology for these measurements themselves.) From those they should be able to show that $T \ne \frac{p^2}{2m}$ in general and that at the highest available potentials $T \propto p$.

*If the device works to low enough energies they may be able to extract a value for $m_e$ under the assumption of special relativity.

1 It may be possible to simple buy a suitable apparatus. I don't know if it will come in under US$250 thought.
A: Say you're driving slowly over a bridge, and you drop something out of the window.
Say there's someone in that river looking up and sees you.
Now what curve shape does that something take on its way down?
To that observer, it's a parabolic curve.
To you in the vehicle, it's a straight line down - air friction aside.
So, the shape is relative to the observer's movement.
