When burying a loved one who was of the Greek Orthodox faith, the standard rites are generally observed. There might be a wake, generally where the Trisagion will be recited for the first time to commemorate the deceased's journey to Heaven. The funeral service will take place at a Greek Orthodox Church or approved funeral home, usually with an open casket, and then it comes time for the burial. If the deceased had expressed an interest in having a cemetery monument at their grave, now is the time to begin discussing it amongst family and friends of the deceased.
The Timeframe for Construction
Because cemetery monuments are made to order and can be robust, often ornate pieces, it's difficult for them to be completed by the time of the burial. Depending on the type of monument desired and the workload of the monument mason, it could be a number of months before the finished product is ready to be installed. You should of course agree to the timeframe before committing to the monument, but remember that it doesn't need to be installed at the time of burial, and often cannot be.
Design Choices
This gives the family of the bereaved some time to come up with the most appropriate design. This will largely depend on your budget, along with deciding upon a style that best commemorates the deceased. You might wish to walk through the cemetery where your loved one is buried, as well as other cemeteries in your area where Greek Orthodox burials have taken place. This gives you an idea of the sizes and styles available, and can be an excellent starting point. If the deceased's spouse wishes to be buried alongside them at a later stage (and a dual burial plot has been purchased), remember that it can be less expensive to buy a larger monument for both, with the details of the spouse to be added when the time comes.
Extras
Once the approximate design has been decided upon, you will need to consider any extras that you think would have held importance to the deceased. They might have wished to have an inscription on their monument, and this text will need to be finalised. If it's a biblical quote in Greek, you must approve the wording and spelling before work commences. If Greek is not your first language, get a native speaker to approve the text. You don't want the deceased's monument to have grammatical errors. You might also like to include a photograph of the deceased, to be printed on a ceramic tile and added to the monument.
You might want to have a small, informal ceremony for family and close friends when the monument is installed, but this is entirely optional. What's important is that the deceased's final resting place now has a monument to commemorate their life and Greek Orthodox faith.