How do I start driving traffic to a low searched google keyword?

Alright, the story is I have a friend who wants to start a website on this new service he's offering, but after checking Adwords for the key words that match his new service, they're very low as almost no one in the field provides what he does. Let's say, some keywords come up with nothing, others with under 100 local monthly searches.

Would I still optimize the website on those keywords, and then have him build some kind of social (on/offline) campaign that targets those keywords to start building them up, or should I try just try to optimize his site on keywords that do not exactly describe his service, but are in a vague range of it?

In this situation, I would first concentrate on low-hanging fruits, that is exactly on keywords that are low volume.
After that, you should expand optimization to more general terms.

Just one curios question, is your friend sure he chose right keywords? You should also scout competition to see what keywords are they using.

That has a really simple answer.
You have to a great seo work to do that in your favor.
If you do not have enough idea about this then you can not do that easily.

I'd try to rank for the less-exact (but more popular) keywords. You can still have pages that are optimized for the specific terms, but if the pie isn't big enough, why fight for the majority of it? It should be easier to rank for related keywords than to convince the general public that they should search for the exact keywords you had in mind, unless it's a new product or service that will become more popular on its own.

You should build some back links on these keywords and then lets see the result yourself!

Have you had a good honest look to see if it is worth the time and effort?
I rank number 1 on page 1 for 2 keywords but they only get about 200 searches a month so it is very, very low. I only did it to prove to a client that it is easy to do but there is no money in it from adsense, lead referral or similar.

You can do it but will you get a return on your time and effort?