Hopefully it's not the All-in-One version -- don't have a lot of experience with them, but I imagine they're almost as difficult to work on as laptops.
So hopefully it's one of the desktop models.
http://h10032.www1.hp.com/ctg/Manual/c02778024.pdf
Bad news is, yes, it's an OEM power supply, so the quality is going to be low.
Good news is...you may have more wattage than you thought. Luckily you don't have that USDT model (130W or 180W?!?) -- basically, the USDT model is limited to the 65W CPUs, & your i5-2400 is a 95W CPU, ergo you have at least the Small Form Factor (SFF) model, or one of the tower models (MT or CMT) -- so you're looking at 240W for the SFF version, or 320W for the MT/CMT models. That's not great, but it should be enough to power a
Of the 4 GPUs you mentioned, 2 of them (GTX 550, HD 6770) are at the same tier (http://www.tomshardware.com/reviews/gpu-hierarchy,4388.html). However, both require too much power; they're both rated at over 100W of TDP, which almost guarantees they need a PCIe power cord (& I can just about guarantee your PSU doesn't have that).
-- SFF can take low-profile cards, but the x16 slot is limited to 25W maximum. I'm not sure that's enough power for those cards in particular. Heck, I'm pretty sure that's enough for a low-profile GTX 750 or 1050. Your options would be more limited to things like a Radeon HD 6450, an R7 250, an R7 240, or (probably your best option) a GT 1030.
-- MT & CMT can take full-height cards. The x16 slot can provide 75W (enough for a GTX 1050 or even the older GTX 750TI), but
only if you don't have anything in the x4 slot; if you do, it cuts off at 35W (apparently the x4 & x16 slots share power). Again, you can't rely on power directly from the PSU, but you could get a GTX 1050 (or, if you can find one that doesn't need PCIe power connectors, a GTX 1050TI).
Now, don't know how powerful these GPU options would be, but since the system is probably running off of the integrated graphics right now, you should see some major improvements.
If you need more performance, consider the following:
-- if it's running a 64-bit version of Windows, then try to max out the RAM. It may not take the fastest RAM, but it does take DDR3, & yours will take up to 16GB (4GB per slot).
-- See if you can find an i7-2700. It's a little faster, & the extra threads can help out.
-- Check the size of the PSU. If it's a standard ATX size, then you could get a better retail PSU. Getting at least a decent-quality 450W, or even 550W, PSU would allow you to put a really decent GPU in there. Unless you're going to also get a new monitor, though, I wouldn't look for anything better than a GTX 1060 or RX 480/580.