One electrician told me that armored cable (BX) wiring must be used in a kitchen island cabinet. Another electrician told me that simple Romex is fine. Which is it? Does the NEC require BX in kitchen islands?
The relevant NEC is 334.15B
(B) Protection from Physical Damage. Cable shall be protected from physical damage where necessary by rigid metal conduit, intermediate metal conduit, electrical metallic tubing, Schedule 80 PVC conduit, Type RTRC marked with the suffic -XW, or other approved means
This is more of a "It's a really good idea if you do this" area (with regards to the island). It depends a LOT on the design of the island itself. I've seen BX wires that were overkill and I've seen NM wires that were unsafe the way they were routed.
What I would do is find an electrician who isn't in the Always BX camp. Look at the design with them and figure out if there's anything that could cause damage to the wire. If it's always going to be in an inside corner, the risk is low enough you could get away with NM (and I would say most islands could). If the wire will have to run tight to, say, metal tracks for a drawer (where it could make contact), then run BX to be safe.
Let's start with the term BX. BX is no longer allowed in electrical wiring due to its poor grounding qualities. MC (Metal Clad) or AC (Armor Clad) is used in it's place as it contains a full size grounding means. I know this may seem trivial but there could some misunderstanding if we don't use the right terminology.
Now let's look at you question from a different angle. What kind of wiring can be used in a kitchen island? This can be found in Chapter 3 of the NEC. Article 320 is for AC, 330 is for MC and 334 is NM. Each article lists the uses permitted and uses not permitted. It's easy now for some like yourself to go online and google these Articles. You will find that the cabling types I mentioned can be used in most situations you are asking about, but there are some exceptions. This is why answering your question is a little tricky since there could be three right answers or one very wrong one.
So when you look at how you will be locating your cable into your kitchen island you can determine which system best suits your needs and if you have a question maybe it should be more like:
If I am run under a sink in a kitchen island, is that considered a wet or damp location and is NM permitted?