The following theories come from the Journal of Joseph William Summerhays:-

"One (theory) is that the family was originally called Summers, but they became so numerous and the given names duplicated so much that a change was deemed necessary. Therefore, when one of the Summers family married a lady by the name of Hawse, they made one name of the two and called the children of this couple Summerhawse. Why this is so I cannot tell as they all come from the same stock.

The other theory in regard to the origin of the name is that during the middle ages, grants of land were given by the Crown of England to persons for services they had rendered. These grants of land were fanced and were called "Hays", "Cot" or "Worthy". It appears that a man by the name of Summers came in possession of one of these "Hays". The name of Summers was coupled with it; hence the name "Summerhays". While I have seen the record of the marriage of Summers with Hawse and know that such a marriage was duly recorded, I do not vouch for either theory. I merely give them and the readers will have to place their own estimate of the value of either of them."

From SurnameDB

"This interesting name is of English topographic origin for someone who lived by an enclosure (on upland pasture) where animals were kept in Summer. The name derives from the Old English pre 7th Century "somer", plus "hoeg", an enclosure. The Medieval English word "hay(e)" from "heog" was used in particular to denote an enclosed forest."