The search bar does a wildcard search on the following fields:

• Product Sku
• Product Name
• Product Summary
• Product Description

Additionally, you filter by category on the fields listed above.

We recently implemented a new Product Search Enhancement that provides the ability to enter search keys for your product that will be searchable from the product listing page search box. This is useful if you would like to have products searchable on words that do not show up in the product name or description or for common misspellings. The new search management screen is accessed from the product setup page to enter additional search keywords. You could use this to enter the product "suggestions" you asked about in your post.