Henry is the intern that worked at Treasure Island before Jake did. As explained in several logs from TGsEE, Henry wandered off without supervision, and Lisa and Greg could not find him. This later led to them hiring a new intern named Jake Smith.
The former creator of the game, Matthew Phoenix Rodriguez, stated that Henry was to make an appearance in the game. It is unknown as of yet how he was to appear, whether through a suit, a phone call, or even as himself.
In Purity's version, Henry would appear in the Night 3 Phone Call. He states that he knows what's going on and why this is all happening. He states he can't tell Jake over the phone, as "They might be listening". He tells the player to meet him in Pirate Caverns, and that he will tell them everything they need to know, but is cut off before he can say anything else.