I'm thinking a PIR sensor (available from many places -
ebay $9.99 link
Adafruit $10 + P&P link )
Together with a microprocessor with a serial interface e.g. ATTiny2313v $1.48 from mouser link. The ATTiny will run with no additional components, using an internal oscillator. You may need a bit more memory (RAM/Flash) than this but even an ATMega168 is not too expensive.
Use a SD card to store the data in (fast and non-volatile) - can be interfaced directly to ATTIny via SPI (need a SD card holder). No need to use FAT as the data can be read out by the controller into the serial interface.
Use the serial interface to plug your laptop into (bring a USB-TTL serial convertor).
So with enclosure and assorted connectors etc you will probably come in at about $20.
The PIR operates at 3.3v (you can bypass the 3.3v regulator to do this) and the ATTiny will work quite happily at that voltage.
Battery consumption is the only other issue here - PIR consumption is the one to worry about here - the microcontroller can run in full sleep mode (micro amps) almost all the time, waking the microcontroller up via an interrupt when the PIR detects something.
The PIR is rated at 4.7mA when run from 12v. You can get rechargable AA cells with around 2000 mAH, or LiPoly cells - so these would last just over two weeks.
So all in all yes it is possible - but it may cost a few $ more than your budget.