I should have enclosed more info sorry. My roommate gave me the hat. His mother was a huge fan and went to a convention the hat was purchased there and then signed by Jon. Here is a bad quality photo for now. Will have a better one
https://www.dropbox.com/sc/r5k5r5rqvdkvsfa/AABDZ_78yeENzC-VyQXDjH9da
Okay, so it's just a regular hat that happens to have DeLancie's signature on it.
This makes the "value" of the object somewhat harder to really pin down. Because now you're talking about a $20 hat signed by an actor, rather than an on-screen prop/piece of wardrobe used on the show signed by an actor connected to that item.
When it comes to any item and determining its value its mostly dependent on how much someone is willing to pay for it and what makes it worth owning, not to mention the rarity of the item.
So, say, you had a prop or wardrobe item from the show signed by a cast member connected to that item. You now have a very rare item signed by an actor. It's going to be worth the value of that item, given that it's rare, unlikely to have a duplicate or many duplicates on the market, so possibly 100s of dollars.
Here? You have a hat. Which, as a hat, is pretty much worth nothing. You have a signature, the value of which is harder to determine. DeLancie does cons so getting his signature isn't *too* hard if he happens to be at a con you can go to, meaning his signature is "worth" $40 or so, usually what you'll pay for a signature at a con.
His signature on that hat? *Maybe* $50, being generous on the rarity of the hat, the likelihood DeLancie's signature is others and how much such an item is going to be "wanted" over something more connected to the actor, character or show. (Like a headshot, or a prop/prop-replica more connected to the character.)
So that'd be my guess, it's worth maybe $50. But like any piece of memorabilia or items like this it's worth as much as someone is willing to pay for it. If someone is willing to pay $100 for it then it's worth $100. If people are only willing to pay $20 for it then it's only worth $20.
If you're interested in selling it there's multiple things you likely need to do. First of all, you need to find a way to get the signature authenticated. I'm sure there's services and such out there that will/can do that and provide documentation verifying the signature but, naturally, that'll cost which will eat into your "profits" on the sale. (It'd be hard to sell the item by upping the price of it to account for the verification.)
Once the signature is verified and you have documentation to back that up you need to find a price to put on it as a "base", the least you're willing to take to part with the item, this base needs to be reasonable and, obviously, cover any costs of the item. (Which now includes whatever you originally paid for it -including the signature- and now the verification.) Make this price your Reserve/Buy it Now option if you sell it at a place like eBay. At the very least you'll learn if this price is too high or not, if the item doesn't sell you know it's too high-priced.
If you meet/beat your reserve or get the Buy It Now price then you make money, minus eBay's commission.
To "make money" off the item this is probably the best and only route to go. If you watch shows like "Pawn Stars" or any other shows of that genre you should know the problem you'll run into if you try and sell the item directly to a retailer. A retailer will low-ball the price the item is worth because he has to buy it and turn around and sell it and make money himself. So selling it to a buyer isn't going to get you anything, as that buyer will be in the same position you are in. He can only sell it for so much to make money, meaning he's only going to pay less than what the item is worth. Which does you no good.
It's a complicated business dealing with items like this, really.
FWIW: I don't think I *personally* would pay very much for the hat as, for me, there's really nothing about it that makes me think of the character, actor, or show beyond the font on the hate and the word(s) on it. For my signed items I have head-shots of the actors and a couple other items that remind me of them/characters they've played. (Like comic books with their likeness on it.) But that's just me. I'd have no way to hang/display a hat and looking at that hat won't make me think, "Gee! John DeLancie!" Besides, for me part of the experience of a signed item is meeting the person and having the item personalized to me. (Making it worthless to anyone else.)
So, that's my thoughts. It *could* have value, but I wouldn't guess it's very much.