Yes, Linn became a common spelling in the late 17th century and even appears that way on some Ayrshire maps of that period. However, the spelling in Ayrshire eventually reverted back to Lynn, the most common spelling in 13th-16th century documents. In Dalry, Ayrshire, seat of the family, modern maps show Lynn Falls, Lynn Glen, Lynn Avenue, etc. My Scottish records go back, in some cases, to the 12th.
It's interesting that you mention Londonderry. One member of the Y-DNA project of which I'm administrator is descended from James Lynn, who was born in Londonderry (a/k/a Derry) in 1790, moved to Knockahollet in County Antrim, and emigrated from there in the early 1850s. His gravestone states specifically that he was born in Derry. Here, he and his family settled in Exeter, Green County, Wisconsin.
In any case, it's important to keep in mind that the way a particular name is spelled should not become a stumbling-block to your research. The Lynns/Linns are a perfect example. You might wonder how I know the Ayrshire family had their name spelled seven different ways. The thing is that they owned several properties and appear in numerous documents concerning those very properties, and have their name spelled different ways at different times. The spelling of names just wasn't consistent in those days.
I hope your research goes well and leads you to interesting things!