RIGR Vol 12 pp 247 & 248 has her maiden name as Jones. At least one online family tree has her name as Penelope Jones (no source noted).
However, I believe she was Penelope Fones, since Daniel Fones Essex is mentioned as a nephew, s/o Benejah Essex, in the will of Daniel Fones who died c1817. Therefore, I have entered her maiden name as Fones.
Also, see this entry from the RI Cemetery Database:
ESSEX PENELOPE ([FONES]) 1765c - 18 JAN 1827 WK054