Providence Island was a Caribbean colony founded by the Earl of Warwick, John Pym, Lord Saye-and-Sele, Lord Brooke, and other future Parliamentary leaders.  If it's remembered at all in history, it's remembered because the Earl of Clarendon identified it as the place where rebellion against the king was organized (and he wasn't wrong, really).  

Plus, it was the first English colony to really embrace slavery.  A number of its settlers would go on to become Americans.  

And it was a spectacular failure.  And, discussing why it failed while places like Saybrook and Virginia succeeded gives us insight into "who Americans are and why," perhaps more than anything we've discussed so far.