Using a plural possessive pronoun for a singular noun?

I feel no closer to the answer.

I feel that there is no real answer: English lacks the precision to express this clearly in the form you are using.

We only have guidance:

Teacher [to students]: "On the trip, students must not stick their head/heads out of the bus window/windows. Students must keep their hand/hands on their knee/knees."

Each student has only one head but the bus has many windows, but one head can only be stuck out of one window at a time.

Each student has two hands and two knees.

The guidance that the teacher is talking to the students individually and simultaneously. Each student hears the words as if they apply to him/her individually and so each sentence is phrased as if the teacher were talking to only one student.

Thus

Teacher [to students]: "On the trip, students must not stick their head out of the bus window. Students must keep their hands on their knees."

And then

The students had a wonderful trip to the firing range where bullets flew over their head/heads.

The students had a wonderful trip to the firing range where bullets flew over their heads. Because the bullets went over many heads and there was no individual reference.

Thus in

How can members change their passwords?

You are addressing all members individually and simultaneously, and “passwords” suggests that each member has more than one password.

How can members change their password?

You are addressing all members individually and simultaneously, and this suggests that each member has one password.

Both of

• How can Members change their password?

• How can members change their passwords?

are correct and natural, and both forms are commonly used. In this use "their" isn't really a plural pronoun, it is effectively singular. Personally, I would favour the second, because there are multiple passwords, one per member, but the other form is fine.

If you want to avoid the use of "singular they" you could instead write

• How can a member change his or her password?

• How can a member change that member's own password?

The meaning is the same, this is a difference of style and preference. English does not have non-gendered singular possessive pronouns. Attempts to coin such a term have not gained wide acceptance. The use of "they" (and related forms such as "their") for a single person of unstated or unknown gender has hundreds of years of history in English, and in recent decades has become significantly more common. It must now be accepted as part of normal usage, whether one likes it or not.

You write "I have read that a plural pronoun should be followed by a plural noun." This is not a general overriding rule. There are cases where this is required by the meaning. For example:

The riders choose their horses.

There are other cases where it is not. For example:

The jurors were taken to their room.

In general, where the noun represents a single thing shared by the group or used by each member of the group, it will be singular, and where there is a group of similar things, often one for each person in a group, a plural noun is normal. But one can find exceptions to that, also. Some nouns, like "pants", always use a plural form, for example (because they were once two separate garments, one for each leg).

Both uses are mainly synonymous. However, to avoid the danger of making people think of more than one password for each user, you can safely use the singular (password).

I don't quite understand your sentence "plural pronoun should be followed by a plural noun". Where have you read that? There are plenty of examples that invalidate it:

Our house, their school, our body

and the list can go one. Certainly, you can play with singulars and plurals to mean different things, but I will not harp on on the subject.