A friend and I were discussing what would be an appropriate word to describe the person that has been in a group for the longest. I suggested the word elder, but that seems to apply the age of the group member is older than the others. The word we are searching for wouldn't imply that they are older, just that they have been in that group for the longest time. Any suggestions for a good word for this?
Otherwise, you can use (most) senior member, as John Lawler suggests, or oldest member if you don't mind a little ambiguity. Eldest can also imply seniority, but it's more often used to specify the age of a family member, so you're right to avoid it.
Tenure refers to the length of a person's term in a given position. So, calling a group member longest-tenured would refer to their time in the group and nothing more.
See, for example, this usage referring to a coach's length of time with a current team:
On Thursday, Charlie Manuel managed his 1,332d game with the Phillies, making him the franchise's longest-tenured manager.
I think a common term for this in Britain would be longest standing member.
Oldest would normally refer to the oldest (eldest) in age. The Senior member would (if used at all), I think, be understood as referring to authority rather than tenure or age. A senior member is most likely to be understood as referring to one of a group of members over a certain age (typically 60 or 65), who may pay reduced subscriptions.
I don't think we would use longest-tenured as an adjective, and, although we would understand, member with the longest tenure, I don't think we would use it in that manner. (See also my comments under the answer with that suggestion.)
The term I tend to hear used for something similar to this is "lifer". This is generally applied when a person was with a group as long as it is possible to be. For example, my church recently gave "lifer" awards to graduating seniors who were members since the infant nursery, and my son's High School put a special mark in the graduation program for "lifers" who attended school in that same district since kindergarten.
Here's an example I found in a recent news story:
Jim Routon's idea of displacement is moving on the other side Pennsylvania Avenue. He's a Southside lifer in OKC, except for years spent at what then was Central State University in Edmond.
Of course the term is also used for criminals serving "life" sentences.