any relay do this?

1K become 47K, 20K become 15K, and cap is 330uF. It work great for the first few days, but then one night I'm leaving office late (around 1am), and then I just found out I can't turn on the low beam anymore. Then I just drive 50 miles back to home with the high beam, and luckily no one flash me and no cop around. Then the low beam work again when I removed the circuit after I back to home.

The base-emitter voltage of 5V isn't the problem, it doesn't see more than 0.7V there. Depending on the relay being used, it could dissipate over 500mW in the transistor as it slowly transitions between off and on. It's going to peak out right at the middle when the relay gets half of the 13.5V and the transistor gets the other half. A standard automotive type relay coil resistance of 88 ohms results in a peak of 0.518W at a 13.5V supply voltage. The slow transition is forcing the transistor to remain in the active region for a long time instead of acting as a switch. A transistor driven on hard and fast to saturation will dissipate very little power, but it's not doing that in this circuit. If the relay is a lower power relay it'd be less of a problem, but even so, it's going to dissipate some heat and it has to go somewhere. It may not sound like much to a transistor rated for 75W, but that figure always assumes that it's properly heatsinked to get rid of the heat. The transistor should be mounted properly here. Try touching it as it operates to see how hot it's getting. Also might see if the relay contacts may have degraded from the slow switching, it's harder on them when high current has to pass through them without the contacts getting full pressure. Higher contact resistance means the contacts heat up more during the transition period.

I have used that transistor to trigger a relay many times. They do not heat up at all even keeping it energized for hours. However I did not try switching this setup many times to see if that would heat it up. I see no reason for it to have failed. I am hoping he has a failed connection somewhere. I guess we have to wait and see what he finds.

It's fine when it's either on or off, the only time it starts to dissipate any heat is during the switching period. The original circuit pictured has a relatively short delay time, so it doesn't linger in that middle area for long. When the circuit was changed to get the long delay times is where it might start to be an issue. If he didn't repeatedly switch it on and off, it should be okay, since it would have a chance to cool back down. It'd take a while for it to heat up to the point where it might be a problem, since it's such a big transistor. Looking back at why he needed to do this, it's probably not any problems from the transistor heating up, since it's to power HID headlights. I don't think he'd be turning it on and off repeatedly while driving along. I'd guess it's more likely to be a relay contact failure because HIDs have a high surge current on startup, and the slow relay switching might result in the contacts burning up. Or maybe just a wire connection came loose like you said.

When I said heat was not going to be an issue, what I meant was it did not need a heat sink.  I did not realize you were going to dip the board in epoxy. 

I would make sure it works for a week or so before I coated it.  Put it in a project box and maybe inside of a plastic bag for a week or two to make sure it is going to hold up.

You could still see if the transistor is working without digging it out of the epoxy, if you can hear the relay click at all, it should still be working okay. You might try to check the relay and connections then.

He already ran a dremel tool through the case of the transistor.