How can I fix a pixel on my LCD monitor?

There are a number of methods said to be effective, but as always, "your mileage will vary". There are primarily two classifications of methods, one is physical - manipulating the pixel by hand, the other is electronic - manipulating the pixel by software.

Physical methods can have undesired effects. Many such methods require applying pointed pressure to the pixel. This process can leave scratches and even holes if done without care. If a user attempts to correct an issue with a product and in the process causes more damage, it certainly will not be covered under warranty so be sure you know the consequences of undesirable outcomes.

Electronic methods are far more benign, but requires turning the pixel on and off again in rapid sequence, effectively causing the individual pixel or pixel area, or the entire screen, to flash. Below is one such manipulation of pixels by software.

Pixel Flash

This file is a composite web page with a animated GIF background. When viewed full-screen in a IE browser, it can be used to correct some pixel faults over time.

Warning: This is not to be used by those prone to seizures caused by flashing lights.

For users who discover one or two defective pixels in a new monitor, the solution is generally to return the monitor to the retailer for exchange. Most LCD monitor manufacturers have a pixel policy that follows the International Standards Organization's specification. The standard specifies the maximum number of kinds of pixel faults (per million for each size of monitor) allowed before the monitor is deemed defective. The warranty therefore would likely follow this standard as well. For end users who cannot exchange, this can be discouraging as one or two defective pixels very likely does not exceed the maximum.

Check out this wikiHow page and this Tom's Hardware article for more information.