Is it possible to change the handlebars on your bike to drop handlebars relitively cheaply?

There's no way of telling, there are so many things that can influence the cost. The things you have to have is of course the bar, bar tape, new brake wires and a pair of brake levers.

What you probably will need is another pair of shifters, and assuming the bike was properly sized to begin with you are also likely to need a new stem.

The cheapest option is to use something called "drop bar ends", and maybe a pair of drop bar brake levers.

Do note that the final outcome of this is very dependent on your bike's basic configuration, merely slapping a drop bar on something like a hybrid or a MTB won't make it a road bike.

People usually run a frame one size smaller when using drop bars for instance. Also note that v-brakes as a rule aren't compatible with drop bar brake levers.