<p>I have an STL file from thingiverse. The model is of a rectangular lid with an engraving. I would like to print it using two different colors, so that the engraving would be in a different color than the lid base. In the model description, the creator explained that he simply switched the material mid printing.</p>

<p>However, I have a two-extruder printer, and I'd like to utilize it for this printing. What's the easiest way (tool) to select a part of the model and define that it should be printed using a different color?</p>

One method is to use meshmixer to select the faces to be created in the second color and "detach" them without removing them from the model. Keeping the detached faces in place provides the appropriate alignment when brought into the slicer. When exporting the model for printing, check your preferences to ensure both segments are exported at the same time. The preferences (file) default to separate exports based on selection.
Some manipulation will be required with the extracted segments, as typically one is removing a zero-thickness surface from a manifold object. The videos I've seen usually create an extrusion towards the inner body to create a dimensional model from the extracted skin.
Even if you are not familiar with Meshmixer, there are a number of videos and tutorials explaining this feature. If you use the terms "meshmixer dual extrusion" you'll get loads of links in return. I narrowed it by adding "Maker's Muse" to the search, as his explanations are clearer than others.
I know that Slic3r and Simplify3D will support the correct positioning when importing the model, other slicers may require additional research to accomplish this objective. My search also appears to indicate that Cura will support such processing.
The rabbit in this image was being prepared for two colors. I've seen the video but cannot recall why this image was presented, as the other pieces were eventually incorporated during one print. It's possible this image represents a single color print, based on the supports, but it is indicative of a partial process during the creation of a dual extrusion print.