I see you've already found the Lennis Family Forum at Genforum:
http://genforum.genealogy.com/lennis/ Unless that was a different Dave who posted there a couple of days ago.
The nearest I can determine, is that "Lennis" may be a variation of the Scots surnames "Lennie" or "Lennox".
According to lexicographers Hanks & Hodges, "Lennie" is a Scots habitation name from "Leny" in the parish of Callander in the former county of Perthshire, so called from the Gaelic word "leana", meaning "bog" or "marsh", and the local suffix of "-ach".
"Lennox" is a Scots and Northern Irish habitation name from the district near Dumbarton, recorded in 1174 in the form of "Leuenaichs", in the following year as "Levenax". It is thought to come from the Gaelic word "leamhan", meaning "elm", and the local suffix of "-ach".
It seems likely to me that your name is a variation of one of the above names, with the "-is" suffix being a form of the typical "-ach" suffix.
-- Nanaea