So turns out there's this concept called "Escheatment" where governments will take custody of unclaimed property, like if a check is sent to your previous address so you never cash it because you didn't know about it.
In the US, this is handled by state governments. If you google "[state] escheatment" you should see a .gov website that will let you search your name to see if they've got anything for you so you can file a claim. There are aggregator searches like missingmoney.com, but it didn't list some of the entries that the official one did so I wouldn't recommend it. You should check any state you've lived, and maybe your family too if you're feeling saucy.
I didn't find anything in my name, but it showed similar results by last name and I found stuff for my mom, dad, sister, a few aunts and uncles, and even my late grandmother! It usually won't be much, but it's your property, so you might as well get it, right? Plus in the case of your family you get to ring them up and say "Hey, how would you like some money?"
You can also claim stuff from a deceased relative but the requirements vary from state to state. Definitely a death certificate, not sure if you'll need a copy of the Letter of Testimatery (sp?) or anything that ID's you as the executor of the estate of the deceased as well.
I don't know if EVERY USA state has their own local listings but where I currently live they seem to run their own.
Edit: Just did a quick check between the site you listed and the site on one of my state's .gov sites and they are.. in fact.. the exact same search results.
I think it's slightly different in how it gets "related" results. When I searched my name on the govt site it gave me results for my dad that the aggregator did not, although I could still get them by searching his name directly.
Good to know.