Google "crawls" websites and stores that information so when someone searches a relevant term appropriate sites are displayed. Since the site is new, it may not have been crawled yet. There are things you can do to make it rank higher.
Set up a Google Places page (google it to see how). Once that is set up, ask his references to review him on Google Places. That links back to your website and since google places is considered a reliable, high value link, it helps your ranking. You can do the same with Yahoo business (even if you don't get reviews on that one) and Yelp. Definitely do yelp - that's a great place to have reviews too and they will email you weekly with your activity. Also Google+.
You also want a facebook page for the business and you want to put the FB link on the website. They make little icons you can use to put on your site so when someone clicks it it takes them right to your page. Get all your friends, family, neighbors...everyone you can...to like the business on facebook. Facebook is also a reliable link so it helps your site to be connected to it. There can be a business Pinterest account too with all his work.
You've heard this one before, but a blog is a great way to rank higher. When your website has a lot of fresh content (versus being a static virtual brochure) it ranks higher. He could do a ton of blog posts on home remodeling issues. For instance, he does a post on "How to Choose a Home Remodeler" and when someone searches for a home remodeler in your city his blog is relevant (which is part of the website). Blog posts can be linked to facebook and people can subscribe to the blog and share the posts...all these things are going to help you tremendously.
I bet in 10 minutes we could come up with over 50 blog post topics relevant to his business. He can also have a youtube channel and do video blogs. Like a quick instructional on how to tile a backsplash or how to create a tile design for your backsplash. You would link your youtube channel to your website too.
You also need to track your website with google analytics. It tells you how much activity your site is getting and where it's coming from...again, just google it to figure out how. It does't make you rank higher but it helps you figure out where your referrals come from etc.
SEO is search engine optimization and it's the art and science of getting your site ranked higher. It looks like weebly has some tips for sites built on their platform. Check out this link and make sure you are optimized. You may need to add meta-tags to photos or something (google can only crawl text so pictures are ignored. By adding tags with key words in them, google picks them up in their crawl) http://weeblyseo.weebly.com/
I know you were hoping for an easy answer but there isn't one! Ranking high is hard work and it will take time, but you can get there eventually. You can pay for ads (pay per click) so you will show up at the top when certain key words are searched on, but eventually you want to be ranking high organically (unpaid). All the interaction between the sight and other sights with authority (facebook, yelp, city search, google +, twitter....) will help. If you want to do PPC ads let me know and we can set up an adwords account for you and get some ads going. I would wait until he is ready to take on more work. Don't really want to run them if he's got a ton of backlog and no time to do it all.
BTW, when we bought our company our website was buried several pages back and now look at us!
Hope this helps! (and isn't too overwhelming).