Anything that takes longer than it should, such as:

Understaffed or officious immigration.
Slow baggage handling.
Lack of available ATMs at international terminals.
Badly situated public transport that takes forever to walk to.
Car rental desks with laborious procedures that means a simple pick-up means 15 minutes of paperwork and upsell (and a queue of people doing the same).
Badly designed airport terminals where the walk from the gate is just silly - this particularly applies to mixed domestic/international arrivals where one is dominant, and the other involves a passenger route along a circuitous corridor up and down steps (LGW north, I'm looking at you...)

Then there's the gouging:

"Official" taxi companies whose rates are astronomical.
Public transport fees that are double (or more) the usual fare for that distance.
Exit-through-the-gift-shop forcing you to carry your luggage through an unnecessarily convoluted arrivals shop.

I would also like to see:

More in the way of decent public showers for use after long-haul arrivals in Y (the ones in NRT are a great example).
More practical shops. Little supermarkets like at AMS and now at UK airports are particularly useful.
Better waiting area for greeters. AKL's "amphitheatre" does this really well, and coffee shops arranged along the arrivals walkway are good as well. But for goodness sake, please direct people away from blocking the only way out for their big emotional reunion.