Is there some deeper meaning to the jar lid specifically, and hope's status of remaining under it?
Absolutely. Hesiod is quite deep, and as Aristotle's use of Hesiod's Eros demonstrates, profound meaning may be found hidden in the work.
And Hesiod says, “ First of all things was Chaos made, and then/Broad-bosomed Earth . . ./And Love, the foremost of immortal beings,” thus implying that there must be in the world some cause to move things and combine them.
"Hesiod... assumed Love or Desire as a first principle in things."
Hesiod has two accounts of the Pandora story, one which mentions a jar (i.e. box) and one that has no mention of any vessel save Pandora herself. See "Good and Evil in the Pandora Story"
The verb ἔμιμνε, which is used to connote the status of Hope under the lid, is and imperfect verb. Imperfect verbs are generally regarded as "open ended" as in continual, which connotes a sense of eternality in the Hesiod passage. [Compare to the aorist or pluperfect which connote completed actions.] The choice of imperfect, in and of itself, is hopeful. "Remained", as it is often translated, might be more properly "has remained" or even "remains".
The pithou of jar Hesiod references are made from clay, and yet Hesiod says this particular vessel, at least the lid, is unbreakable.
Continuing with the body metaphor, the "lid" can be taken to mean the head.
Hope, by contrast, relates to the mind.
The unbreakable status of the lid can be taken as a metaphor for the eternal component of humans, which is often called the soul.
Thus the evils of the Pandora's box may destroy the body and bring death, but they cannot destroy the human spirit.