Science-fiction writers are often credited with "inventing the future" with some of their ideas. For example, the idea of geostationary satellites is often attributed to Arthur C. Clarke.
What novel was the first to mention or predict a personal handheld computer that anyone could use?
Jerry Pournelle, in several TWiT podcasts, claims that he and co-writer Larry Niven came up with the idea in The Mote in God's Eye published in 1974, however, I'm sure an idea like it came along well before that.
Isaac Asimov's 1958 short story "The Feeling of Power" posits a population completely dependent on their "pocket computer" for doing basic arithmetic. However, they are not described in use for anything other than arithmetic (which is, after all what the big boxes did in 1958), so I don't know if it counts or not.
The way the characters use the things in the Niven/Pournelle reference more closely resemble the things we think of as PDAs/smartphones/netbooks.
It looks like Asimov has him beat, but Arthur C. Clarke mentioned an electronic news pad in his 2001: A Space Odyssey. And in 1980, Allen MacNeill predicted that by 2010, we would have handheld computers (that would be connected to a mainframe).
In his novel The Age of the Pussyfoot (1966/1969‡), Frederik Pohl describes a device called a "joymaker"; a scepter-like device that is connected to a central network and functions as a voice-operated computer.
‡: The Age of the Pussyfoot was first published as a novel in 1969, but before that it was published as a serial in Galaxy Science Fiction starting in 1966.
In computer science, Alan Kay's Dynabook was first described by him in 1968. It was a pad style computer with a keyboard and LCD screen, similar to, but smaller then, the Kindle. I don't recall any science fiction stories using a hand-held computer prior to that.
A. E. van Vogt's fixup novel The Mixed Men (1952) has the following passage:
There was no whine of sirens, so it was not a battle alert. He put
down his book, slipped into his coat, and headed for astrogation and
instrument room. Several officers, including the ship's executive
astrogational officer, were already there when he arrived. They nodded
to him, rather curtly, but that was usual. He sat down at his desk,
and took out of his pocket the tool of his trade: a slide rule with
a radio attachment which connected it with the nearest--in this case
the ship's--mechanical brain.
John Brunner's novel, The Shockwave Rider (1967), had protagonist Nicky Haflinger hacking the "datanet" from a folding pocket videophone (which required a wire connection to the telephone system to operate).
Though one could argue that it, and even the home data terminals almost everyone had, weren't primarily computing devices in their own right, none the less they gave anyone with appropriate skills (in the case of the pocket veephone, the ability to program in phone code groups) nearly unlimited (given the ubiquity of phone connection points) access to nearly unlimited computing power.