What would be the disadvantages of giving a company or product a one-letter name?

Hard to find in google or other indexes / directories / listings.  A lot of forms will reject the name as being too short, thinking it is an error.  One registration difficulty might be with fictitious names and  secretaries of state - corporate paperwork.  You'll probably end up having to adopt a longer formal name, like "O Pharmaceuticals, Inc." In addition to formal trademark difficulties (run into others, hard to gain for yourself), what's behind trademark -- branding -- is difficult because the field is crowded, and because the letters themselves already have such strong connotations that making them represent your brand will be hard, like painting a mountainside.  Think of O.  Who is there.  Oprah, Oxygen Network, http://Overstock.com.  The connotation with some brands is "the big O"...  How could you fit in with that?  You have to be a strong brand to try to take that on, and more likely you like they will just be using O as a contraction for branding purposes, after you've already become established.

It would be impossible to find on search and impossible to trademark. You're going to run into conflicts with stock tickers too. It's a far more practical idea for a product name than a company name, but probably a bad idea regardless. Aside from the legal concerns, there are marketing ones as well. A one letter name is very hard to bring to market. The only type of company who could pull this off would be a massive one with an equally massive budget. Motorola tried this with the 'Q' a while back. Sure, a one letter name is ubiquitous, but that's not really the point of a good name. You want to stand out, not fit in with a million other things in a million other markets.