What would be the most economical way to run wire for a driveway light?
Your two choices if you want normal AC power out there are:


Direct burial cable. Figure something like $0.50/foot for 12 gauge cable, though it's possible that you will need to buy a roll of 500' instead of just 300' since 250' or 500' are standard lengths. If you do this, you will need to make sure that it is buried 24", and I highly recommend putting a bright plastic tape at 12" so anybody who digs will hit that before the cable. You will need to use conduit to protect the cable where it comes up to the surface. 
Wire in conduit. PVC conduit is the simplest, and you are talking around $50 in conduit. You would then need three wires; two insulated for hot and neutral and a bare one for ground. Something like $0.10/foot for each 12 gauge wire these days. You will also need at a couple of intermediate boxes; you can probably pull cable through 100' of conduit at once if it's mostly straight, but 300' is stretching it, so you'd pull in sections. If you do this, pull all 3 conductors at once and use cable lube to make them pull easier. Pro tip: tie a piece of paper to a string, stuff it in the conduit, and then use your shop vac to pull it to the other end. This gets you the pull string simply and easily. PVC conduit doesn't have to be buried as deep (18" IIRC). 


In either case, you are probably going to want to rent a trencher to get that deep. 

Now, to figure out the size of the wire you need, and to do that, we need to know the resistance of the wire you might use. Looking at the AWG gauge table, we see the following values:

14 gauge copper has a resistance of 2.5 ohms per 1000'
12 gauge copper has a resistance of 1.6 ohms per 1000'
10 gauge copper has a resistance of 1.0 ohms per 1000'

You only have 600' of wire, so multiple each of those values by 0.6, to get

14 gauge - 1.5 ohms
12 gauge - 1.0 ohms
10 gauge - 0.6 ohms

We can now calculate how much voltage we will lose for your 300' run at each of the gauges I listed. This is where things get a little hazy; electrical code does not mandate a maximum voltage drop, it's more a matter of what the equipment on the other end can tolerate. Lights may get dimmer and motors can work to hard and die if there is too much voltage drop.

Code does recommend that you don't exceed a 5% voltage drop, so I'll use that for calculations. 5% of 120 volts is 6 volts, and using ohms law (in this case, current = voltage / resistance), we get the maximum current we could pull and stay at 5% voltage loss:

14 gauge = 6 / 1.5 = 4.0 amps
12 gauge = 6 / 1.0 = 6.0 amps
10 gauge = 6 / 0.6 = 10.0 amps

Multiplying by 120 volts gives us wattages:

14 gauge = 480 watts
12 gauge = 720 watts
10 gauge = 1200 watts

So, if you need 480 watts or less, you would be fine running 14 gauge wire, and so on for the other gauges I list. If you need more than 1200 watts, you're going to need to go with a bigger wire, but I'm pretty sure you don't want to buy 300' of 8 gauge cable; it will be pricey.

If you are going with lighting at the other end, I highly recommend looking at LEDs. Not only do you get a lot of light per watt, most LEDs should retain brightness even with a 5% voltage drop.

If you are going to all this trouble, I recommend putting a standard outlet in a waterproof box out there. You won't be able to pull too much power through it, but it could be very handy. I'd also think about whether you would want anything else out there (intercom? video? other sensors?) - if so, it's much easier to put extra cables in the trench you are going to dig.

Hope that helps.