I have a picture of a friend however I've lost contact with them and can't find them by name. Is there a way to use this picture to find their Facebook profile?
Have you tried Google Image search?
Facebook profile photos are public (for discoverable accounts). If they have used the photo relatively recently a reverse image search may be helpful.
From Google Image Search, click the camera icon in the search bar ("search by image"), then choose the option to upload an image. (If you have a modern browser, you can also drag-and-drop an image file.)
Google will then use the image to find other images that either match exactly or come close.
There isn't a way within Facebook to use reverse image searches.
There is no official way to do so.
But if you found that picture from web and have not changed the name that file, you can search the id of Facebook profile using the number given in photo file name.
For example: suppose you have downloaded that pic from web and it saved as 41480_10738811_7375_n.jpg, so here the middle number i.e. 10738811 is the profile ID from a Facebook user's profile. At the end put the profile ID number you copied from the filename: http://www.facebook.com/people/@/10738811
I personally have not tested this. You can give a try.
I am afraid that the answer given by serenesat earlier does not apply any more.
There are other alternatives as well apart from using AI E. solution of Google images.
Keep in mind that you cannot upload an image at Google Images’ mobile site since there’s no option to upload an image. You can do search though with the help of some text.
You can also use other "Reverse Image Search Engines" such as:
FindFace could do the trick, but I ignore if is still available
You should try.