I've done a project using Google Search and another using Solr so I can help you to choose:
Google Search
- Your site need to be online somewhere so that google bot can index it.
- Your SEO should be good enough.
- The index is built automatically based on page rule (meta name "robot").
- You can provide a sitemap with all pages you want to index.
- Nice backend where you can manage the indexation.
- You need to learn the API a bit if you want to customize search pages layout.
- It's not a free service (starting at 100$ per year for 20 000 queries).
Solr
- You have to index your site yourself (but as you control your indexing you have a better control of the content indexed.)
- Then you have to keep the index up-to-date.
- Easy to customize (you can display search result as you want).
- Nice feature included (suggestions, auto-completion,...)
- It's free.
To sum up if you are on a tight schedule Google site search can be faster to implement at first but Solr is much more customizable and free.