How much does it cost to restring an acoustic guitar?

Very little- the only cost is in the strings which can be as low as £5 . You can buy these off the internet (they tend to be cheapest off sites like amazon) and restring it yourself which is really very simple (i'll lave instructions at the bottom of this answer). Or you can buy the strings from a shop, and if you ask a lot of shops will restring it for you there and then, at no extra charge. If they do charge you it should be no more than a couple of pounds/ dollars. One final option is to get a kindly friend to show you how to do it- which should defo be free :P

So if you do try to restring it yourself......

Actually not as simple as "taking them off and putting on new ones".
When replacing strings on any instrument, you want to do them one at a time. So, for example, on a guitar, I start with the low E string (low meaning tone, not position). Remove it by unwinding it until it's loose (never just cut strings) and then replacing it. Then move to the next and so on.
The reason why we do this is because there is a balance between the string tension and the neck tension. If you remove all of that string tension at once, it can "mess with" the natural memory of the wood and metal truss rod that make up the neck.

In order to actually change the strings you need to take the new string and hook it into the bridge (there should be a small lump which hooks in.
The take the other end and poke it through a small hole which should be in the bolt on the machine head- leaving about 5/6 inches on the other side. This is the fiddly bit so you might need to try this next bit a couple of times.
OK- so pull both ends of the string against the bolt so they bend and stay in place. Then start turning the machine head (you can do it in either direction- just ensure you turn them all the same way). Whilst doing this you need to keep pulling on the end which is attached to the bridge in order to keep the tension and get it wound nice and tight. Keep turning- this may take a while until it is fairly close to the right tension.
Something I would advise is running your finger under the string when it's nearly tight enough which stretches the string slightly so the guitar won't fall out of tune so often after changing.
Repeat with all the other strings until VOILA! you have a newly strung guitar.