I don't know if it qualify as home experiment, but you can use the internet to get access to thousands of kilometres of optical fibres for free. It allows you to measure a lower bound for the speed of light in the fibres, which is $c/n$, where $n$ is the refractive index of glass, typically around 1.5. This corresponds to $2\times 10^8 \text{m/s}$. Using ping, you measure a round trip time, that is it should correspond to 100 km/ms of round trip.
From Paris, I ping the website of Columbia, in New-York, I have
fred@sanduleak2:~$ ping www.columbia.edu
PING www.columbia.akadns.net (128.59.48.24) 56(84) bytes of data.
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=1 ttl=113 time=125 ms
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=2 ttl=113 time=116 ms
....
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=16 ttl=113 time=112 ms
^C
--- www.columbia.akadns.net ping statistics ---
17 packets transmitted, 16 received, 5% packet loss, time 16023ms
rtt min/avg/max/mdev = 108.585/118.151/132.156/7.728 ms
The minimum round trip time is 108 ms, which would correspond to 10,800 km instead of 5839 km. Off by a factor of 2, but the correct order of magnitude, due to delays in switches etc., which is why we said this is a lower bound.
If one looks more precisely the trajectory of my packets to New York with tracepath
fred@sanduleak2:~$ tracepath www.columbia.edu
1: sanduleak2 0.266ms pmtu 1500
....
3: pioneer.ens-cachan.fr 1.072ms
....
6: vl172-orsay-rtr-021.noc.renater.fr 28.747ms asymm 9
7: te0-1-0-5-paris1-rtr-001.noc.renater.fr 20.931ms
8: renater.rt1.par.fr.geant2.net 30.307ms asymm 9
9: so-3-0-0.rt1.lon.uk.geant2.net 33.780ms asymm 10
10: so-2-0-0.rt1.ams.nl.geant2.net 36.570ms asymm 11
11: xe-2-3-0.102.rtr.newy32aoa.net.internet2.edu 127.394ms asymm 12
12: nyc-7600-internet2-newy.nysernet.net 128.238ms
13: columbia.nyc-7600.nysernet.net 135.948ms
14: ....
We see that the packets travel around (Paris, London, Amsterdam) and cross the Atlantic between Amsterdam (10) and New-York (11) in 127-37=90 ms (roundtrip). This still gives us a 9000 km distance, way too long. I don't know if it is due to the cable trajectory, electronic delays, to small sampling by tracepath or an error on my calculation.
Related to this ping delay, you have the funny 500 miles bug.
Another in-the-lab experiment using cheap material and computers is in the arXiv paper speed of light measurement using ping. However, their measurement is indirect (they measure the propagation inside CAT5 cables), but it should also be doable with optical fibres.
Edited to add: My idea of using tracepath probably comes from Measuring the Earth with Traceroute. In this paper they are more lucky than I was (only 20% slower, instead of 100% !)
Best Answer
Simplest thing coming to my mind -- wrapping your mobile (or anything that is able to receive a signal) in some aluminium foil (or in anything made of conductive material).
Then your reciever won't be able to receive the signal. "'Cause conductors block e.m. waves."