0.5 to 4.0 hours Timing Circuit
Since you don't want to use a microcontroller, you might as well take a look at the ubiquitous 555 timer chip.  Here is a circuit that, when activated, will keep the relay energized for a period determined by the capacitor and two resistors on the left side of the schematic:



The 555 timer only triggers when it's input goes from a high level to low level.  I don't know which way your PIR is set up.  The nice thing about this circuit is that it includes the circuitry to trip on a negative going pulse, or a positive going pulse using the transistor to invert the signal.  The transistor, if needed, can be any NPN type, such as the 2N3904.

With the values shown (where the resistance can vary from 4.3K to 104K depending on the pot setting), the time period is adjustable from 17 minutes to 7 hours.  I figured that using this 555 monostable circuit calculator.

The capacitor is what's known as a "supercap", here is one on Digi-Key.  220mF is the same as 220,000 µF. Note the tolerance is -20% to +80%, so you'll have to play around with the pot to get the right setting.  But even at the extremes of the two values, you should be able to set your desired range.

The TLC555 can only source a maximum of 15 mA, so you will want to use a reed relay.  This one will operate off of 5V and draw 10 mA.  Note however it is a SPST type (it doesn't have the NC contact like the one in the schematic.)