Unfortunately, Rick, I have several Lydia Eveleths born in that timeframe and none of them born in MA and no middle names of any of them. Let me give you the three I have and perhaps something will click:
Lydia Eveleth b. 16 May 1802 to David & Deborah (Nightingale) Eveleth in Athens Township, Windham Co., VT
Lydia Eveleth b. 28 May 1802 to Joab & Lydia (Gowing) Eveleth in Jaffrey, Cheshire Co., NH. She married Abraham Stanley 17 Apr 1832.
Lydia Eveleth b. 12 Jan 1804 to Jonas & Abigail (Parsons) Eveleth in New Gloucester, Cumberland Co., ME.
Let me know if any of these are of interest to you and I'll be happy to give you what info I have on their lines. I can be reached directly at Sharon.L.Sabol@dartmouth.edu.