Common English Surnames ending in S

Solution 1:

According to Mckinley's A History of British Surnames, the major rise of surnames derived from a personal name with the addition of -s or -es was among the "peasant" classes in the late 13th century. These people didn't have hereditary surnames of their own, so the implication is that they adopted their master's name as befitted their social status as they were bonded to him. If you follow that logic, it primarily signified whose property they were as much as anything, although the passage in that link doesn't wholly explore this.

There was then a second wave of Welsh -s names that appeared more widespread in the British Isles in the 16th century. Together with the earlier English 'mutation', that generated most of the -s names we see today.

So they're not quite patronyms, then, but close.

Solution 2:

One of the better places to look is with The Guild of One Name Studies a web-site used by genealogists etc.

I couldn't find Peters (it appears not to be one of the names for which they supply a possible etymology). However one that falls into the category is Phillips which you will note from the etymology they deem to be patronymic.