The obvious answer to this question is: saying "cheese" makes us bare our teeth in a smiley type of way. But there are plenty of words that do that! "Teeth," "lean," "velveteen" -- basically anything that makes that long E sound. Why did we pick "cheese"?
The idea is that the "ch-" in "cheese" positions our lips in a way that better facilitates “an automatic smile,” according to one of the first references of the trick in The Big Spring Herald in the 1940s. Former president Franklin D. Roosevelt is thought to be the one to suggest "cheese."
Of course, saying "cheese" didn’t even matter until the 1900s. During the Victorian era, you wouldn’t really smile for photos at all. A small, tight-lipped smile was considered more fashionable and evidence of good etiquette, and photographers would often tell their subjects to say "prunes" to get that pucker just so. Plus, it used to take so long for photos to be snapped back then -- the "prune" smile would have been much easier to hold than the big "cheese" grin.
Smiles for photos started moving past children, peasants and drunks with the dawn of the film industry. Once we started seeing movie stars and politicians smiling on film, it transferred to photos; suddenly, it wasn’t as daunting to be captured smiling frivolously. Cameras started getting speedier, so it was easier to maintain a grin.
"Cheese" may be arbitrary, but hey: If it ain’t broke, don’t fix it, right?