I will answer for the timer version. If I could get a proper datasheet for the PIR sensor, I might answer for that also.
Since you only have 3V, you need to use a CMOS 555 timer, AKA 7555. All cap values are microfarads.
The current draw will mostly be in the IC, 60 uA typical, 200 uA max. AA batteries have a capacity of about 2000 mAh. In standby, the circuit will run for 2000/0.2 = 10000 hours or over a year. Motors typically draw considerable current, so if people are pushing the button often, it will last much less.
I built the circuit. The current draw in standby mode is 82 uA. My ICM7555 is from Digikey. If you buy from a non-reputable source, you could get a counterfit part and all bets are off.
Note that simple brushed motors are extremely electrically noisy. It can be extremely difficult to get a circuit to work on a solderless breadboard. The noisy right side must be physically separate from the 7555 circuit. The battery and motor wires should connect near the MOSFET and/or flyback diode. I had to add a snubber across the motor to get my circuit to marginally work on a solderless breadboard.
Edit: I built the circuit on a soldered perfboard. It works better, but the electrical noise is still affecting the timing. It is 5% lower when the motor is running.
A 1.5 Volt alkaline battery will fairly quickly go down to 1.2 Volts. This is the value that I attempted to design to.
The circuit works down to 2.4 Volts. But this is really outside the guaranteed range of the MOSFET. MOSFETs designed for 3.3V logic are common, those designed for lower gate drive are not common. For a more reliable circuit, remove R3 and power the left half with 3 AA batteries.
The snubber RC (R6/C5) is optional, I did not install it on the soldered version.
You should use some care in the layout. Since you are inexperienced, you may want to follow mine closely.
Or, you might look for a delay module. It is unlikely that you will find one that will work on 3V. This would work on a 9V battery. They don't specify the standby current, so I don't know if this will last an hour or a week on a single 9V battery. You will still need a separate 3V for the motor.
This would be a nice module for my stash, so I am going to order one for myself. I will publish the standby current here.
Edit: I received this module, standby current is 15 mA, not suitable for your application.
https://www.amazon.com/DROK-Controller-Delay-off-0-01s-9999mins-Adjustable/dp/B07RGT5G1X