If you want an LCD display - forget it, the cost of a custom LCD and zebra strip connectors would be prohibitive unless you were making these in 1,000+ quantities.
Without a display and provided you don't want a very loud beep, it is just about possible. I would suggest a 3V CR3032 type battery, a disc buzzer and a simple spring switch. Use a small PIC, maybe a 10F200 which in SMD package only has 6 pins, leave it powered up from the battery all the time and use the button to reset it. The software would then run a timer routine for the required time then it would drive anti-phase signals to the disc buzzer so it had ~6V of AC across it. After buzzing it could go to 'sleep' mode and draw almost nothing from the battery until reset again.
That uses one pin for reset, two for the buzzer and leaves one which you could use to select alternate beep pattern or maybe two different pre-set time intervals.
Brian.