Several years ago we were issued an airport designation number by the FAA. The information was lost when my father died. How do I find out what our designation number is?
What you're looking for is the FAA Airport/Facility Directory (A/FD). You can search it on the FAA A/FD Advanced search page. If your runway is registered as an airport with the FAA, its airport identifier will be in the A/FD. It will also include the runway numbers (there will be one for each end of the runway,) as well as other information about the runway (surface type, exact heading, altitude, etc.) See How are runways numbered? for more information on runway numbers. Another website available for searching A/FD information is airnav.com.
Another way to find information on the airport is to look at a Sectional Chart. For most private airports, an 'R' with a circle around it will appear where your runway is located and it will have the name of the airport beside it. You can then use that name to look it up in the A/FD. Skyvector.com is an easy way to look at sectional charts. It works a lot like Google Maps, except that it shows you the FAA sectional chart for your area rather than a normal map. You can just zoom in on your area to find the airport marker.
Your airport will look something like the one that I've circled in red on this sectional chart image:
If your runway has a hard surface (i.e. it's paved with asphalt or concrete) and is more than 1,500 feet long, it might look like the airport circled in the picture below rather than having the circled 'R'. In this picture, the purple circle indicates the airport and the white line within the circle indicates the location and orientation of the runway.
Search directly on the FAA page.