How can I find a death record with little information?

Do what you can to narrow down the dates. This can mean scouring through census information in Scotland and Ireland to find her parents' details (if you can). This might also be in parish books.

Same with Claire. If she came to the US and was alive during a year ending in 0, then she should be somewhere in the census. That will also tell her age at the time, and the ages of her parents if they were alive at the time, and any siblings alive then, too.

Think creatively and go with multiple spellings. Even if you know for certain that it's Claire, there may be records where it's misspelled as Clair or Clare.

Also try find a grave and check with https://www.genealogy.com/ and https://www.ancestry.com You may have to pay some fees to get access to the more detailed information.

Go in with a family member for ancestry.com -- for me, it was one of my third cousins and he mentioned I think on Facebook that he was filling in a tree. He made me an editor and now, between the 2 of us, we have something like 50 people listed. That can also be a way to link with other trees and you may find cousins you didn't know about. They might have better records or some details you can use.

Good luck; it's a fascinating topic.