Mostly to add my vote to suggestions that have already been made:
- Walk-through U-Boat - both inside and on deck. And fill the U-boat with more crew doing routine things and reacting to battle/damage stations.
- More interesting campaign missions. Something better than "Patrol <some random grid>" over and over (mines, agent-drops, cargo-hauls, timed rendezvous, etc.).
- Better crew management. Be able to draw up some set rosters and have them automatically change/rotate if left alone.
- Make success/failure count. Does a single boat make a difference? Well it does if you can engage the imagination of the Führer to divert money into the U-Waffe, or if you let your boat (with Enigma machine and code books) be captured.
- Multiplayer-vs-multiplayer. And escort group vs a wolf-pack would be awesome.
- False alarms. Sometimes the crew get edgy and over-react. A shadow or a cloud is interpretted as a contact, and the boat crash dives.
What we DO NOT need... (these suggestions have been made, but I disagree)
- Sub-vs-sub. A single torpedo and it's game-over. (This is why there are no SQUIDs in SHIII - the Dev's even said so in their interviews).
- Underwater marine life. Maybe hearing distant whales would be nice, but don't waste development effort for something that you never would have seen. The seagulls are cute, but we don't need full models of turtles/sharks/jellyfish/etc.. If it makes the ocean lonely, well... it sort of was.
PS: Sorry for using Atlantic Theatre terms, but I just don't know anything about the Pacific, and I figure the experts will know what I mean.