Interestingly, much of Jewish burial rituals are custom and not halahah in the traditional sense. Although something might be uncommon in one Jewish community it is normative in another. For example, a receiving line is not a normative Jewish practice during a funeral. In Philadelphia, however, you will often see a receiving line after a Jewish funeral at the local Jewish funeral homes. Each community establishes its own norms for burial, taharah (the cleansing ritual), and monuments. Particularly in Jewish communities with smaller communities you will often see a wider net being cast in order to ensure that everyone’s practice will be included in some way. And fortunately, with few exceptions Judaism is flexible enough to allow for most needs to be met.
To answer your specific question, it is quite normative to have etchings on a headstone, but not of the person. The most common image is a star of david with the letters pey”nun written inside. These letters represent the phrase “po niftar – here lies buried.” However, images of the deceased are strictly forbidden in Jewish law.
The Hatam Sofer offered a famous ruling on this very issue. His concern was that engraving a picture of the deceased was close to idolatry and would be inappropriate. Particularly during his time, when it was normative to see someone praying beside the grave of an ancestor, this practice would have been as though the person was praying toward an idolatrous image. Additionally, the picture of the deceased would violate the biblical prohibition of graven images.
Although each community and each cemetery association forms their own regulations and normative practices, it is certainly uncommon to find images of the deceased on the monument. In my opinion, we should also take in to consideration the needs of the community and the needs of the individual family of mourners. Perhaps there is some compelling reason to include an image of the deceased. In that case, if it did not violate the practice of the burial society or cemetery association, I would certainly consider it.
Our job is to give Jewish tradition and practice a vote in our values-based decision-making process. I find the traditional monument to be a meaningful practice which connects each generation to our ancestors and treats all Jews, regardless of wealth or circumstance, equally. Because of that and its place in Jewish tradition, I would be hard pressed to allow the practice you describe.