Let it run for a bit. If you're getting the HP logo that generally means that the motherboard has bootstrapped the CPU, there's at least a chance that what it's doing is a full memory check and that if you wait 10-20 minutes it will eventually POST and boot like normal. I grant it's not a very good chance, but considering all you have to do is just leave it running for say a half hour, it doesn't cost you much to try.
Also, the CPU you have is an Ivy Bridge. It's 4xxx for Haswell and Haswell Refresh, 3xxx for Ivy Bridge, 2xxx for Sandy Bridge, and 8xx/9xx for the first gen, not that it matters a great deal in this case. The IGP is on the CPU, so if you're getting any kind of display at all it means that the CPU and IGP have been bootstrapped successfully and the failure is in initializing some other bit of hardware.
So you should also try removing all USB devices except the keyboard and mouse, which should be plugged into the computer directly via the rear ports (at least for testing). If you've got a spare keyboard and mouse you can use for testing, even better. Some computers are just more sensitive to the slightest defects in USB devices. I have an old computer which booted up just fine with a couple external HDDs plugged in. New one, with the exact same drives, will sit there for 2-3 minutes if the HDDs are plugged in. So, if you can test with other devices, even better.
If you're still not getting anywhere after that, remove any internal components not needed for POST. So RAM and CPU since your video card is on the CPU. If you get far enough to say there's no OS or to insert some kind of boot media, you can shut down and add the HDD back to the system. Rinse and repeat until either everything works again or you find the problem component.
Anything I can do at home to tell which component is bad? Does it sound like the motherboard? How can I test it?
HP Pavilion p7-1420t PC
Intel Core i5-3330 3.0G 6M 77W CPU
6GB DDR3-1600 DIMM (4+2) RAM
Intel HD Graphics (Sandy Bridge)
Window 8.1 64-bit
My 12-yr old son reported that computer would not start while I was traveling last week. The computer is shut down every night. Last Wednesday, a black screen with blue HP logo was as far as it would get during startup. I looked at it today. The computer powers up and the hard drive spins for about 1 second with the yellow hard drive light illuminated then fading. No beeps. The fans continue to run. The wired keyboard does not seem to have power as no key lock lights illuminate. I tried all USB ports. I tried start up with the F2, F8 F12, delete and esc keys pressed independtly and together, anyway, with no effect. I also tried a wireless keyboard in all USB ports with no effect either. The keybords work when attached to a laptop. The HP logo comes up instantly as the computer is powered up but that is the extent of the startup. The monitor is a lenovo so its not the monitor screen. I have an image of the screen if there is a way to post it. I have also pulled and reset the RAM and removed and replaced the BIOS battery. USB ports not hot. This computer has given us no problems other than being Win 8 OS...lol. It has never blue screened. I have Norton 360 which runs every 3 days because of the kids using it while I travel and I run Windows Defender once monthly. There is no recent history of viruses or any application or other downloads or any external/internal component additions recently. All significant software is up to date.