Generally typeahead/autocomplete is instant, and to avoid confusion I'd recommend sticking with the standard of autocompleting instantly after a single character.
For no reason other than convention I'd suggest keeping an instant, one character autocomplete; every search bar I have within reach acts this way, including Google.com, Google Chrome, iTunes, Visual Studio, SQL Server Management Studio, Google Search on Android...and if Microsoft, Apple and Google all agree on something, they often have a reason. Not that you should blindly follow convention, but conventions are comfortable and low friction so you generally need good reasons to start breaking them.
Any delay you add risks making users question whether it's working; is it really an autocompelte field? It autocompleted before, why isn't it now?.
Depending on how (and how well) your autocomplete works, users might actually expect results to be instant or to come after one letter; after months of work with Chrome I've learned what website it will bring up when I type "t" (twitter) "g" (gmail) or u (ux.stackexchange). If I had to wait or type an extra letter it would slow that workflow down. As long as the autocomplete results are not automatically inserted, meaning you have to manually clear them, there's little danger in being a bit overzealous.
However if you have something more than just autocomplete, like Google Instant's search, you might insert a bit of delay to avoid swapping out large areas of the screen (or performing slow/data intensive operations) repeatedly during ordinary searches. Google Instant waits for maybe a few hundred milliseconds before showing "instant" results, and Google Search History only records events where you stopped typing for three seconds. Three seconds is probably overly conservative in most cases, but it's interesting to note that the more "important" events (autocomplete, then display results, then log as history) each take considerably longer. Autocomplete is not as annoying and has little downside, so it displays instantly. Displaying results requires more of a screen refresh and more network traffic, so it's a bit slower. Logging the history is more "permanent" so to avoid clutter/misleading results the timer is quite conservative.