So my mother, before she killed herself, may have told me some things about some bit of equipment that was really, really, classified.
At this point, I doubt it’s a state secret anymore, but how do I find out if it’s okay to talk about without getting myself in deep shit if I’m wrong?
IANAL, but I have held clearances for decades. If you don’t hold a clearance, being given classified information will not get you in any legal trouble. Your source can get in trouble, but that doesn’t sound like much of a problem here.
If you reveal the information and are subject to an investigation (not likely, unless it’s pretty major [e.g., where the moon landings were staged, or how they faked the Kennedy assassination]), you may be ordered to reveal your source. Do so, and basically no sweat (ala Robert Novak in the Plame leak); refuse and maybe get in hot water (Matt Cooper, same case).
I guess if you really want to be thorough, you could put in a FOIA request for the info and see if it’s still classified.
That car that doesn’t run out of petrol?
From the other side of things (not having any kind of clearance, but working with confidential client information on a daily basis): if it’s big, juicy, exciting information, the downside of telling and then finding out it’s classified is outweighed by the excitement of telling. On the other hand, if it’s not exciting information, there may be just as much of a downside for telling, because some people are kind of fussy about things they think of as secure, so you could end up in just as much trouble for telling a much more boring story.
Bottom line, everybody’s got secrets. Some bigger than others, some more important than others. Telling other peoples’ secrets has a limited scope of appeal, whereas keeping other peoples’ secrets lasts forever.
Let’s say I was curious whether a ‘black project’ has been declassified. How could I make a FOIA request without divulging the nature of the project and/or the codeword? 
Not saying this is the solution you’re looking for, but if you just want to talk about without getting in trouble, one way is to reveal it anonymously. Then, once the secret is out in the open, you can publicly talk away without problems (at least, as long as you don’t say so much that it reveals you were the anonymous revealer).
Of course, this assumes that you either think the info should be revealed (whistleblowing) or don’t care.
My advice–if telling won’t definitely make the world a better place…say nothing.
Discretion is a virtue.
If your mother had classified data, then presumably she worked with some agency or organization dealing with that data. Ask one of her old co-workers.
If it’s serious enough, you could be debriefed in the event of an inadvertent disclosure. During that debriefing you’d be compelled to sign some sort of non-disclosure agreement which would then put your balls on the line if you talked about it. Not being a lawyer though, I don’t know if compelled means “no choice” or you’re a spy or something. They’d appeal to your sense of patriotism first I suppose.
I’ve worked with lots of classified equipment that is outdated now. Still technically can’t talk about it, but no one would be interested. That’s not my decision to make though.
Chronos is right too. You can always call the organization and ask to speak with the security manager. Tell them you might have classified. That might get you interviewed, and asked to sign something. Might get you laughed at.
The problem with asking the co-workers is they aren’t supposed to confirm or deny, but if it really isn’t a big deal, they’ll probably say so.