The solution (that can use almost any camera or camcorder) is relatively easy, since the Record Start/Stop function is typically a "toggle" (push-on, push-off) using the same button on the camera. (To save typing, from here on I will say "camera" to mean either a digital SLR in video mode or a real camcorder unit.)
Whether using a wired control signal (for cameras with LANC or Visca remote plugs or DSLR shutter cables) or an infrared emitter connected to a laptop, PDA (remember the PalmPilot, Treo, & iPAQ?), or "universal remote" app for cameras with IR remote control features, a simple 1-minute timer device or program/app can be set to toggle ON (whether by the person pressing a large "Start" button, stepping on a floor sensor switch, or triggering a motion detector) then the timer will trigger another "pulse" when the counter reaches 60-seconds, to toggle the function "OFF".
(I believe iPhones and most smartphones still have IR emitters that can be programmed as "universal remotes", which would be an easy way to test your setup using the InfraRed control method.)
Many cameras already have intervalometers (time-lapse start/stop timers) built-in, and 3rd-party external intervalometers are fairly easy to find, both off-the-shelf and D-I-Y. Just do a Yahoo!Search or Google the keywords "camera intervalometer" and you'll find plenty of YouTube Do-It-Yourself examples and electronic "widgets" for sale that could be adapted for your project.
Newer cameras with HDMI connections can also be start/stop controlled over the HDMI interface connection. (I'll link to a previous Y!A Answer of mine discussing those details. Look in my Sources below, or in the Comments section, if you close-out the Q&A early.)
hope this helps,
Hidden camera & remote controlled camera experience for TV commercials.
Electronics hobbyist over 40 years.
Wikipedia article on LANC / Control-L wired remote protocol: http://en.wikipedia.org/wiki/LANC