Here's an article on how it's made and the ingredients commonly used: https://luisafanzani.com/how-to-make-a-liquid-lipstick/. Getting all of the proportions right is part art, part science when starting from scratch. If it's dried out, it may not be just the solvent that's evaporated, in which case adding solvent wouldn't get it back to the same thing. Also, some of the ingredients may need to be incorporated in a certain sequence, so just adding solvent to the dried-out material may not yield a usable result. I suspect, though, it is a bit like house paint in the sense that once it dries it changes, and you can't just redissolve it and use it.
But if you could, you need a way to blend everything, which is hard enough when you start with the liquid ingredients. The solidified material would probably need to be finely ground as a first step. It's made with a high-shear mixer, and you may not be able to replicate the uniform consistency without it. Trying to hand-stir one bottle's worth isn't likely to do it, especially if you just add some solvent to the bottle of solidified contents.
The usual solvent is isododecane, which probably isn't a local, off-the-shelf item. There are a few alternatives, but they aren't any more readily available. You can get it online (the linked article mentions
makingcosmetics.com in the comments), but the ingredients are typically sold in quantities used for manufacturing.
Liquid lipstick is designed to be long-lasting. There are products sold for removing it. It may seem like one of those should dissolve it, so you could just mix a small amount into the dried bottle to liquefy it again. However, the removers that do dissolve it (as opposed to just breaking the bond with your lips), also do other things that destroy its properties. So it probably wouldn't be usable, or at least very long-lasting.
So the bottom line, you will probably need to call the bottle a loss.