That neon light flashes pretty fast. If you don't have a super high traffic website you should try to leave it on for at least a second so you don't damage the cathode or blow out the transformer.
According to the Google Store that light is around $28 bucks, meaning you're probably not dealing with the highest quality components (read: fire hazard).
The schematic may be incorrect, as it seems unlikely that the relay would energise in an emitter follower configuration, where it'd have approximately 2.6 volts across its coil in the "on" state. Perhaps the relay should be between 5v and the collector of the transistor. Some relays are sensitive enough, though, so I'm not sure.
Relay connected to 5v with the diode across the coil. The base of the transistor goes to pin 18 via a resistor. The transistor is on the other side of the relay between it and ground.
That makes more sense to me because the diode is now actually protecting the Raspberry Pi 5V line.
The schematic is wrong, the relay and diode should be moved to be 'on top' of the transistor. But the circuit on the perfboard looks correct.
The diode goes across the relay coil because when the relay coil is turned off, there will be an inductive spike generated. The diode protects the transistor.
Where did you get your light? I love the idea of a light made into a word. If you had a light with the name of the website on it, you could have multiple websites with lights flickering to show when people were visiting each one. That'd be cool.
UPDATE: If anybody is interested, it turns out you can special order these things with custom words on them. Looks like a great idea for promotional items.
Tried this out with a few LEDs on a cobbler. Works fine for a couple of hours and then ceases to flash.
No crashing, error reporting. Has this worked for more than a day for you?