Hi,
First of all Ron - oops, you're right....that'll teach me for not checking...I've got so many different 555's laying around, I just wack them in a circuit and see what happens
Well, as you're switching 2.4-3.4 AMPs, then that is the reason the schematic uses a MOSFET

You see standard bipolar resistors are nice and easy to use, but they develop a voltage drop across the emitter and collector (when they are 'switched on'). At 3 amps, this voltage is likely to be quite high, like 0.8V, whichc means the transistor is 'absorbing' 0.8 x 3 = 2.4 Watts, and will get very hot without a heatsink.
MOSFET's on the otherhand, have a 'on resistance' when they conduct. For the MOSFET you're using that is extremely low, around 0.05ohms. Using the equation P = I^2 R (power equals current squared, multiplied by resistance) it will disapate 3^2 x 0.05 = 0.45.
In the data sheet you provided, that gate voltage thresshold is called
VGS, and states this is a maximum of 2.0V. This is the minimum voltage needed to 'turn on' the MOSFET so it conducts. Once you're over this voltage at the gate, it will conduct, and the more you go over this threshold voltage, the lower the 'on resistance'. (RDS in the datasheet).
If you look on the table,
RDS, which is 'static drain-to-source on-resistance' it shows the resistance of the MOSFET for different gate voltages and drain to source voltages/currents. The lower the gate voltage, the higher the resistance, but...it is still pretty small....60mohms (yes, mili ohms).
So.....ultimately, forget what I said about using a transistor, the MOSFET you have is
perfect for that application, and will happily work in that circuit. With a 3.7V supply, the gate voltage will be well....3.7V. So it WILL turn on the MOSFET (its above the 2.0v threshold), but perhaps it will give the MOSFET a slightly higher 'on' resistance.
Sorry for a long and drawn out explaination...I guess I could have said: 'Its fine, the MOSFET is a good idea, and it will work at 3.7V as long as you use a CMOS 555 timer instead of a standard one. - but you wouldn't have learned anything!
Blueteeth