Hi I was facing this problem from long back.After login that remember password prompts too quick to click on save button.
After long debugging I found that in my home page(after login) there is "change password" form was there in code,Initially its hidden but when we click on "change password" it show as popup.When page found successful login it fills that form internally because this form content 3 type="password" fields.Because of auto-fill property of browser. That's why remember password prompts too quick.
Initially I key those 3 password field as type="text" but when user click on "change password" link , internally it changes their type from text to password.
When user successful login to home page, browser doesn't find any password field so that it doesn't auto-fill any data that's why it remember password box doesn't disappears too quick it remain showing on page unless we hide it.