How to Run 12v 1A Electromagnet From Arduino (3v) Without External Power Supply
If your battery can provide about 5-6A at 3.7V then you can use a boost converter to increase the voltage to 12V.  

It would have to be capable of providing your desired 1A output current as a bare minimum.

Basically your coil requires 12W (P=VI = 12V × 1A = 12W).  You have to have a power source, at whatever voltage, that can provide that amount of power.  No amount of magic will get 12W out of a power source that is only capable of, say, 5W.

Good batteries for this are Lithium Ion and their cousins.  They can provide high currents for short periods of time.

A second option is to use multiple batteries in series.  Four 3.7V batteries in series will provide 14.8V.  Three would be 11.1V.  If your coil will operate well enough at 11.1V then three would be ideal.  You could take a tap off the + terminal of the lowest battery to power the Arduino so you don't get any losses from an inefficient regulator.

A further third option, if the coil only needs to be energised occasionally, would be to charge a capacitor up over time with a higher voltage.  It basically uses a boost converter but with much less current. A capacitor is slowly charged to a higher voltage, then the entire charge is dumped into the coil in one go.  This is basically how camera flash units work (the whine you hear is the capacitor charging).