This matter has been resolved by Mudcatter 'Taconicus.' I'm going to copy his post from 2013 right here. He shows how to find the original sheet music. I've followed his footsteps and done it, and the flower's name is 'aronatus.'

I'm not going to post the lyrics again. You can find them in the DT using the first link way above. Here's Taconicus' post:
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Subject: RE: Origins: Wildwood Flower / I'll Twine 'Mid the ...
From: Taconicus
Date: 30 Sep 13 - 05:25 PM

At long last, I've received a copy of the actual published sheet music to I'll Twine 'Mid the Ringlets. It pretty much matches what Genie wrote above in 2001. Here are the actual lyrics, verbatim (including original italics, punctuation, and capitalization):

[see the DT - leeneia]

- Taken from published sheet music in the Department of Special Collections, Stanford University Libraries: I'll Twine 'Mid the Ringlets. Words by Maud Irving. Music by J.P. Webster. Copyrighted and Published (1860, 1862) by H.M. Higgins, 117 Randolph Street, Chicago. Pearson, engraver. (The "1860, 1862" signifies that this is from an 1862 compilation published by H.M. Higgins entitled "WESTERN GEMS: SONGS COMPOSED BY J.P. WEBSTER." The page showing I'll Twine 'Mid the Ringlets shows the 1860 date.) I'd post a scan online but Stanford requires permission for further reproduction.
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Why track it down?

1. There's a glow you get when you are (almost) looking at the honest-to-gosh, true original. This is the closest you will get without travelling to California.

2.   You can get the melody right. The original melody is more interesting than what's going around now.

3.   To learn Webster's charming introduction to the song.

4.   To see the 1860's style accompaniment.

5.    To see the lyrics for yourself.