Most of the developed beach areas of Weh have a small selection of simple places where you can eat. There seems to be very few fancy places, and it often seemed like everywhere I went had more or less the same menu of Indonesian foods along with simple western dishes like sandwiches and breakfast. However, some places did a better job than others, so it was a matter of trying various restaurants until we found one that hit the spot for us.
Tempe Bolon - Indonesian tempe (soybeans) in a spicy tomato sauce with rice
Since many places get no more than a handful of guests, service can be a bit slow. If you want to be served quickly, in many cases it's better to place your order well before you're ready to eat. Scuba divers will often order their lunch before making their mid-morning dive.
One important thing to note is that there is a very limited amount of alcohol available on the island. Some resorts do stock beer, and maybe even something harder, but not much of it. Many of the small family-run food stalls do not sell beer or any other alcohol at all. Pulau Weh is definitely not a party place.
Something you will find plenty of is Aceh coffee. The locally produced beans are brewed the old fashioned way, with a large fabric strainer full of ground coffee soaking in a pot of water. When you order a coffee, the cone is pulled out of the pot and held over the cup. A little more hot water is poured in, and the strong coffee is allowed to drain from the filter into the cup. It's very strong and usually requires a little sugar to go down. Note that there doesn't seem to be any milk readily available in most places, so if you ask for milk you'll usually get a can of sweetened condensed milk.