Timer Switch (Repeat)
This sounds too perfect for the Arduino environment. You've already found that you can't buy a commercial product that does what you want. Yes, you could prepare a schematic and circuit with components like a 555, but why?

You might as well buy a $5 Arduino clone, download the Arduino environment, and have at it. Write in code what you want, in (almost) English, rather than designing an electronic variant.

Of course the issue is switching a (I assume) mains-supply light. There are relay shields for the Arduino for less than $10 - I think you can easily get what you want:

const byte relayPin = 4;

void setup() {
    pinMode(relayPin, OUTPUT);
} // setup()

void loop() {
    digitalWrite(relayPin, HIGH);
    for (unsigned i=0;i&lt;10;++i) { // Ten minutes
       delay(60 * 1000); // Wait for a minute (in milliseconds)
    } // for
    digitalWrite(relayPin, LOW);
    delay(5 * 1000); // Wait for 5 seconds (in milliseconds)
} // loop()