In a technical sense, the verb and the pronoun do make it unambiguous. The trouble is that the unusual name risks sending readers up a garden path, as it were; if you start them off reading the sentence with a wrong interpretation of what’s going on, they then have mentally backtrack and fix it. You don’t want that kind of stumbling block anywhere in your writing, but particularly not in the first sentences.
There are all kinds of possible solutions to this. One is to put the most important verb closer to the name, choosing a verb that droplets of water couldn’t possibly do but a human could. For instance, do you need “sat at his desk” at all? Most people reading about a person staring at a computer screen would picture that person at his desk by default, so maybe you can skip that verb and get on to the next one:
* Rain stared at his computer screen and reviewed his last sentence in his mind.
Another possibility is to tweak the spelling of the name so it can’t be confused with the noun:
* Rainn stared at his computer screen and reviewed his last sentence in his mind.
Another possibility is to use his full name, to make it immediately clear that you’re talking about a person:
* Rain Stephenson stared at his computer screen and reviewed his last sentence in his mind.
Those are just a few ideas, and they don’t even include the most obvious suggestion of changing the name (which I’m guessing would be more painful to you than any of these others! I get very attached to my characters’ names, for better or worse). There are always innumerable different ways of expressing the idea you want to express, and when readers indicate even the slightest bit of confusion—again, particularly in an opening, when you’re trying to grab readers and keep them, and not give them any reason to set down your story and do something else—it’s always worth pausing to consider some of those alternatives.