It's a Hohner 1040, produced in the 30s or 40s before they received the standard black and gold "pokerwork" style pattern. I'd keep it personally, but that's up to you. To get it playing perfectly again will take a bit of work - all the reed plates will need to be removed, cleaned and given new valves, and then waxed back into place with fresh wax. Then they'll need to be tuned. Aside from the reeds, the bellows gaskets could probably do with being replaced, and, assuming the thumb loop is missing, you'd need to get yourself a pair of strap brackets and a shoulder strap.
Your options are to sell this one and buy one in better condition - I don't know what the market is for 1040s in the USA, but given its current condition probably less than $100. Assuming you want to play diatonic, your options are probably limited to restored Hohners, which are much better value for money than new Chinese instruments, and in the UK tend to start at around £300ish for a one row and £500 for a two row. Of course, what type of melodeon you get revolves around what music you want to play.
Option two is to keep this one and play it as it is. If it is in decent shape like you mentioned, it should be a lot of fun. I actually just bought myself a 1040 from a folk festival bargain table, and despite being in much worse condition than yours, it's great fun!
Option three is to keep it and restore it. You've found an instrument that's pretty mechanically simple and potentially very easy to restore, compared to other boxes. And pre-WWII Hohners are very good instruments so, once you're finished with it, it will actually be a good box with a bit of value, rather than a mediocre or poor one. Accordion Revival is the go-to website for piano accordion repair advice, but melodeon.net is invaluable for anyone working with melodeons. You could also pay someone else to do it for you!
Of course, you could also do options 2 or 3 and buy another box, which isn't a bad idea. you rarely find melodeonists who only have one! There's also option 4, which is to sell it and give up, but that's boring.
Good luck!