Say you discover a new element in a far future sci-fi world like the one I'm building. What cool futuristic, scientific function could this element have? Or what could this element help do? I already have a cool name for it. I just need a purpose for it.
If we assume that the element is going to fit into the standard periodic table that exists in our world, the answer is: not very much. If you look at a periodic table, you should notice that there aren't any gaps in it. There are only endless blank spaces as we continue towards heavier and heavier elements. The problem is that such elements are inherently unstable and decay extremely rapidly. For example, Flerovium, element 114, has a half-life of 17 days.
At best, you could get an element that is somewhat stable (i.e. has a lifetime of a few days) and decays into useful daughter products, like Uranium, Plutonium, or rare-earth metals, as well as some energy.
But since any element past uranium will have a relatively short lifetime, you won't be able to find it naturally. On our own planet, all elements past Uranium have already decayed, and we can only make them in the lab. So you would have to synthesize this element directly, using a lot of energy to do so.
Best-case-scenario, you have an element that has a lifetime of a few days and releases a large amount of energy when it decays. You could use this as an heat weapon, or as a sort of capacitor for quick-release energy storage. However, since it doesn't last very long, you would not be able to use it for any long-term energy storage. If it decays very energetically,there would also be issues containing it.
To be honest: not much at all.
Unfortunately for your element but fortunately for chemistry groups in the periodic table tend to exhibit similar chemistry. So your new element would have nothing new to contribute other than the fact that it was new and the novelty would make it interesting, but predictable.
Now: it could be sort of like mercury which has recently be found to exhibit some interesting relativistic effects (warning: YouTube Video) but it is highly unlikely to grant any interesting chemical effects and would like be unstable per Phiteros's answer.
For a much more complete answer see this answer. There are no gaps between 1 and 118 (possibly 119) that cannot be assumed to be outside the realm of chemistry.