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The story is often recounted that the planet Neptune was discovered because of its gravitational influence on the planet Uranus. However, where the idea came from is less frequently retold (why, I don't know). The polymath Mary Fairfax Somerville described the situation before this discovery in her science popularization, The Connexion of the Physical Sciences, first published in 1834:
The tables of Jupiter and Saturn agree almost perfectly with modern observation; those of Uranus, however, are already defective, probably, because the discovery of that planet in 1781 is too recent to admit of much precision in the determination of its motions, or that possibly it may be subject to disturbances from some unseen planet revolving about the sun beyond the present boundaries of our system. If, after a lapse of years, the tables formed from a combination of numerous observations should be still inadequate to represent the motions of Uranus, the discrepancies may reveal the existence, nay even the mass and orbit of a body placed for ever beyond the sphere of vision.
To coin a phrase, forevermore is shorter than before: John Couch Adams read that passage in the sixth edition (1842) of Somerville's book and was inspired to work out "the mass and orbit" of the body in question.
As Somerville described in a later edition:
The honour of this admirable effort of genius is shared by Mr. Adams and M. Le Verrier, who, independently of each other, arrived at these wonderful results. Mr. Adams had determined the mass and apparent diameter of Neptune, with all the circumstances of its motion, eight months before M. Le Verrier had terminated his results, and had also pointed out the exact spot where the planet would be found; but the astronomers neglected to look for it till M. Le Verrier had published his researches, and assigned its place to Dr. Galle, at Berlin, who found it the very first night he looked for it, and then it was evident that it would have been seen in the place Mr. Adams had assigned to it eight months before it had been looked for. So closely did the results of these two great mathematicians agree.
Even professional scientists can learn something from good popularization, now and then.
EDIT TO ADD: Adams wrote a note to himself in 1841, to investigate "the irregularities in the motion of Uranus [...] in order to find whether they may be attributed to the action of an undiscovered planet beyond it"; however, the relevant passage in Somerville's Connexion appears as early as the 1836 edition. (Google Books has it p. 74 of the 1840.) At the very least, it seems Adams was spurred on by Somerville's suggestion.
One reason why we don't teach all the ins and outs of how scientific discoveries happen is that the paths to a discovery can be incredibly twisted and complicated when compared with the discovery itself!
