In Germany there are many places with names similar to Hayde – Hayd, Heid, Heide, Heidt, being some of them. The surnames are prolific in southern German areas around the Palatinate.
I have spent numerous hours trying to trace a Hayde (and variants) from the Palatine to England, Ireland and America in order to try to find a connection between the Irish and German Haydes.
In 1709/10 some 821 Palatine families were sent to Ireland from London, after they had left the Palatine and made their way to London. They were essentially protestants who fled the Palatine due to the persecutions at the time. I however have not found a Hayde or variant in any of these records.
What I have traced is a family who used Hayde (and most of the variants) back to Nuertingen in Baden Wuerttemburg in the early 1500s. This is a very comprehensive tree and can be viewed as Tree # 13 at https://www.ancestry.co.uk/family-tree/tree/51542132/family
There is also a web site for this tree at http://www.genealogy.com/ftm/h/a/y/Karl-Hayde-VIC/WEBSITE-0001/UHP-Index.html
Any further information is welcome.