Question

In a cathode ray tube, why is the filament made of tungsten?

Answer 1

Because tungsten has a high melting point. This allows it to be heated red/white hot without melting. This high temperature results in the thermionic emission (boiling off) of electrons which are needed for the CRT to work.
Tungsten also has a low vapour pressure at high temperatures, so it won't evaporate too quickly, and it is relatively cheap.