the options:
1) a 555 timer based one. but the timing for 6minutes ON may not be accurate.
2) a digital counter based one. you have to wire the counter ICs and make a driver and PCB.
3) a microcontroller based. write a program and with relay driver , 6min ON and 5 sec OFF can be done. You must be already familier with programming microcntroller.
The last option is flexible , with a ready made MCU kit and making a small PCB for driver and you can test with your varying requirement.
Depending upon your controller knowledge(AVR/8051) , you can choose the one that suits you.