TL;DR: "Computer" will likely become an implementation detail, like "transistor" is to a computer today.
Historically, "computer" referred to a person carrying out computations. Like if we called a saw a "carpenter".
Of course a carpenter does more than just saw, so we separate the job description from the tools employed. Human computers, however, literally just did computation, so the name fit the (machine) computer equally well.
As computers have taken on more and more roles, they've just taken over the name of whatever they replaced, though not always a thoroughly as with the original computers.
I suspect the word "computer" itself won't be replaced with any single word, but rather that it'll become a word like "transistor" or "capacitor" - an implementation detail.
For instance, tablets are a pretty new category of computer. It's a computer, but I don't think many would call a tablet a computer in day-to-day speech. Sure, you can point at a tablet and say "that's a computer", and people would agree, but if you asked someone "have you seen my computer?" they'd probably think you're looking for a laptop or desktop PC.
Even the fact that we can use "laptop" interchangeably with "computer" hints that the idea of a computer is less important than the form factor. On the other end, we have mainframes and servers - still computers, but they have their own nomenclature regardless.
The same works for phones, gaming consoles, and other gadgets. Obviously they're computers, but we don't call them that. We also distinguish Mac from PC (which is even stranger, considering both fit the description "personal computer", but that's a historical quirk).
Heck, you can point to washing machines, cars, and thermostats and say they're computers - and you'd be right, but more people would wonder what you mean. It just fades into the background.
One could also take the co-opting human job descriptions further: If a computer is making trades on the stock market (as they they've just done millions of times in the past couple of seconds), aren't they, in effect, "traders"?