Since you are asking about replacing tungsten, and not about replacing tungsten-based bulbs, I will limit my answer to "incandescent lightbulbs", which are lightbulbs that produce light by passing a current through a thin conductive filament (wire), thereby heating it.

The filament gets "red-hot" or "white-hot", thus providing the light. From this, you can see that a possible material for a lightbulb filament should be a conductor and be able to withstand heat - in other words, it has to have a high melting point. The very first incandescent light used platinum for a filament, which is a metal with a high melting point - but also more expensive than gold... The first commercial bulbs used carbon fibers as filaments. Other metals (osmium, tantalum, molybdenum) have been used as well, but tungsten seems to outperform them, at least once production cost is considered as well. On a side note, incandescent light bulbs are slowly being replaced with more energy efficient alternatives, alternatives which don't take the "detour" of heat production to generate light from electricity and thereby use less energy per light output.