Traditionally, stroboscopes were built with xenon flashlights such as those used in photographic flashguns. However, these flashlights require a 400-600 V voltage and even more for igniting them. Needless to say, electric shocks of such voltages are often lethal. Unless you have experience in building such high voltage circuits (including proper insulation) and also have measuring equipment to help with the process, you shouldn't even think of building one.
Fortunately, with the advances of contemporary semiconductor technology, there is no need for those flashlights. There are very high intensity LED diodes available which are perfectly suited for this task. These mostly AlInGaP (Aluminum-Indium-Gallium Phosphate) LEDs come with an intensity level in the range of 10,000 to 25,000 microcandelas, which is an awful lot compared to the 5-50 mcd value of common LEDs. If you even put it into a socket with a reflector and a focusing lens, its light will be so bright that—although it's not a laser diode—you should never look directly into the light rays. Such LEDs are about ten times more expensive than regular ones but in absolute figures that's not so terrible at all: around 3-5 EUR each.