I waited before responding to you, thinking you would be best getting a response from another American Elyard, but since they have been silent I thought I would offer what I could.
I'm in Australia, so my research on the US lines has been restricted. Fortunately, I have had some assistance from a couple of Elyard descendents there, and have managed to compile a fair bit of info.
When I was younger, family members had plenty of ideas where Elyard came from, and I think they covered almost every European country EXCEPT England, which ironically turns out to be the one. The Elyard (Eliard) line is definitely traceable back to the mid 1500s in southeastern England, the counties of Sussex and Kent. From here, they spread to London proper and Norfolk, before heading overseas.
If you haven't come across my research website, you can find it here http://elyard.q.fmhttp://elyard.q.fm Although I haven't made any changes in the past year, my research is still ongoing, although I have pretty much done all I can at this stage on the US lines.
To complicate this line, there seems to be a lot of confusion of surname changes, including a very confusing mix of Elyard & Ellinger, which is still yet to be sorted out with any certainty. The line dates back to one couple, Jacob Elyard and Susan Trent, but their exact origins remain a mystery. Jacob was most likely not born in the US, since had he been it would be likely to find other Elyard lines from any brothers/sisters. The lack of such tends to imply he emigrated, but did he marry before or after emigration, we don't know.
Here in Australia, I have very limited resources for American immigration records, so have not been able to go any further.
To further confuse the issue, the various 1850-1880 US cenuses online also mention a small family of Elyard's in southern states, but the real confusion here is that they are noted as being black. This small family is not included in the tree on my pages at this stage, since I have absolutely no idea how they fit in (yet).
I hope there is something useful for you on my site. The best place to begin would be trying to locate the arrival of Jacob circa1800-1820ish (or a birth record for him if in the US).