Here's a nice simple circuit that should do what you want.
The 555 is connected as a monostable. When you trigger it by closing the switch connected to pin 2, its output (pin 3) goes high (to +12V) and about a second later, returns low.
The time is proportional to the product of RT and CT. Actually it would be better to use different values from the ones on that diagram:
RT use a 1 megohm trimpot (aka preset potentiometer). You can adjust this with a screwdriver to get the exact delay you need.
CT use a 2.2 uF capacitor with a tolerance of 5% or better. Something like
http://www.digikey.com/product-detail/en/C1206C225J8RACTU/399-9333-1-ND/3522851 (multi-layer ceramic surface-mount part)
http://www.digikey.com/scripts/dksearch/dksus.dll?vendor=0&keywords=CWR29JB225JCBC (tantalum surface-mount part)
http://www.digikey.com/product-detail/en/R82CC4220AA70J/399-6027-ND/2704681 (probably your best option)
Each time the 555's output changes state, C3 charges or discharges, generating a spike of voltage across the relay coil as shown on the waveform diagram. This spike is about 20 ms wide from the start until the point where it drops below about 2~3V, which is roughly the point where the relay will drop out.
If the relay stays on too long, decrease C3, and vice versa.
The relay I've specified is a very sensitive type, with a coil resistance of 2 kilohms and a coil current of only 6 mA.
http://www.digikey.com/scripts/dksearch/dksus.dll?vendor=0&keywords=306-1292-nd. You could use a different reed relay with a higher coil current but you will need to increase C3 in proportion to the increase in current, to get the same pulse duration.
The reed relay must NOT have an internal diode in it. (Some reed relays have diodes built in.) This is because the coil is energised with both polarities in this design. If you use a relay with a diode across the coil, it will only activate in one direction.
Also, add a decoupling capacitor (another 4.7 uF electrolytic will do) across the power rails near the 555.