The i5-2400 will work with pretty much any socket 1155 motherboard, so you can look at the boards that were released with SandyBridge CPUs — for example, H61, H67, Z68 — as well as those that were released for IvyBridge CPUs, such as B75, H77, Z77.
Most board manufacturers have a CPU compatibility list for each board, so if you find a particular board, you should be able to confirm by searching for the make/model.
Given that they're not made any more, finding a "good" replacement might be tricky. As long as the motherboard has all the ports and sockets you need, it should do the job. They all at least have a PCIe x16 slot for your GPU. Some low-end boards only have 2 RAM slots — if you have more sticks than that it would be a problem. Most DDR3 RAM will work with any board, but you can also try and confirm RAM compatibility either with the motherboard or RAM manufacturer's compatibility lists. Other things that you need — like the number of USB ports; fan headers; PCI/PCIe slot combinations — and what sized board your case will fit might lead you to accept or reject a replacement.
(If you had an overclockable 'K' CPU it would be more difficult, because the Z68 or Z77 are the best choice there).
2nd hand is most likely a cost-effective option if you can find one. Some socket 1155 boards are still available at online stores, so you could check Newegg, Amazon, and similar places. Alternatively, you could call a few local computers stores to see if by chance they've got old stock.
As an example of the process: here's a listing for the Gigabyte B75M-D3H at Amazon. The CPU support list shows that the i5-2400 is supported. It has 4 RAM slots, and after the GPU is in will have a PCI and PCIe slot free. It is only micro-ATX sized, so smaller than your current board, but most cases that support ATX will fit a micro-ATX board too.
If you change the make and model of the board, you may have difficulty booting your OS with the updated board. This depends what OS you're using, and how different the replacement board is as far as drivers go. Windows traditionally hates hardware being replaced, and may refuse to boot; if you were using, say, Linux it would most likely just work. So keep that in mind that some recovery or worst-case reinstall of the OS might be required. This might lead you to try and match the chipsets as @Paul mentioned.
As an aside, I assume you've confirmed it's certainly the motherboard, and not something else like a dead power supply, that's causing your problems?