Charge the battery with the white wire. Use a bridge rectifier from radio shack it is like $2.50. The rectifier has four "lugs" the diagram is on the back of the package. One of the two AC lugs comes off your white wire, the other AC lug is your engine "ground". Hook the + DC lug to the + battery and the - to the negative battery. Battery charges when the engine is running.
I have seen two sizes for emergency lighting 6v SLA the cheapest of the two also happens to be the smallest 4.2 aH I believe it is slightly smaller than a regular 6v lantern battery. I suppse you could use 4 rechargable AA or AAA batteries to keep it small and compact, but my understanding is that the charging characteristics of any available rechargable battery does not work well with inconsistent and fluctuating output of the white wire unless you want to build a complex charging circuit. An SLA is quite forgiving in that aspect
The problem with running directly off of the white wire is that the output varies greatly with with the engine RPMs and that can't be good for small electronics, the constant change in current. Do a search for the output graph of the white wire, somebody mapped it out, and the easiest way to smooth it out I would think is with a battery.