Normally, a search for
john should find all relevant matches in your contacts, that is, all contacts containing a word starting with "john" in any field. E.g. "John Mills", "Johnny Appleseed", "Elton John", "123 Johnson Lane", etc.
(I think Google Contacts expects names to be entered like "John Mills", not "Mills, John", but searching for "Mills" should work either way. Android Contacts has options to sort by last name and to display last name first. I don't see those in web contacts currently.)
But if you're using the new "Contacts Preview" instead of the older "Contacts" feature, then a search for
mills will show some matches from your contacts along with random matches from Google+. The irrelevant matches crowd out the relevant ones if you have many "Mills" family members in your contacts!
One fix for that is to switch back to the old Contacts:
- On the left-hand tab, click More.
- Click Leave the Contacts preview.
Alternative: People report that typing your query (
mills) then hitting Enter (or clicking the magnifying glass search icon) will get better search results than just typing the query and looking at instant search results. [I can't test it because I left the Contacts Preview.]