If I were to melt igneous rock (for example granite), and then let it cool, would the re-hardened material still be granite?

There are many types of igneous rock. The mineral they crystallize into is determined by the rate of cooling and the pressure it is under.

Lava rock that is found near active and inactive volcanoes, is not granite. It was formed directly from the magma cooling fairly quickly. Obsidian is formed when magma is cooled so quickly that it doesn't crystallize (it's a glass).

Granite is actually a plutonic rock meaning that it formed more slowly in the subsurface as opposed to a volcanic rock which solidifies (forms) at the surface. In nature, igeous rocks are often melted and reformed as metamorphic rocks such as gneiss and schist. Over long periods igneous (and other types of rocks) exposed to extreme pressures and temperatures during mountain building will gradually develop into metamorphic rocks. In an artificial environment such as a laboratory, an igneous rock once melted and then cooled rapidly would likely form a different crystalline structure because granites cool and form very slowly in the subsurface.