In Romeo and Juliet, what does Mercutio mean when he says that dreams "are the children of an idle brain"?
It is important to read what Mercutio says about dreams in context. Note how this quote comes after his long speech about Queen Mab, in which Mercutio uses fanciful description, humorous puns, and mockery to make the point that dreams are not significant. This is of course in response to Romeo's dream that he mentions to his friends. Thus, the quote you have identified is part of Mercutio's concluding argument as he tries to reassure his friend Romeo, to note pay too much attention to his dreams:
True, I talk of dreams;
Which are the children of an idle brain,
Begot of nothing but vain fantasy;
Which is as thin of subtance as the air,
And more inconstant than the wind, who woos
Even now the frozen bosom of the North
And, being angered, puffs away from thence,
Turning his side to the dewdropping South.
Ironically of course, Mercutio is actually wrong. Romeo feels that they should not attend the party, but because of Mercutio's persuasion and reassurance that dreams are not to be paid attention too, they do go, which sets in motion the plot for the rest of the play as Romeo meets his Juliet.