I have a friend named Chloe (pronounced Clo-ee). She writes her name using an accent mark over the e. A friend of ours thinks that the accent mark means unstressed, so that her name is pronounced Clo-eh. Does the accent mark mean stressed or unstressed? I've seen other names that use a double dot over the e to mark something as stressed, like in the name Zoe.
No, Chloë and Zoë are still stressed on the penult even when written as Chloé and Zoé.
Once upon a time people would write learnéd instead of learnèd, but it doesn’t change the stress in English — as opposed to in inglés, where it does.
Standard Written English doesn't use diacritics,¹ so there is no consensus for what they mean when you do use them. That is, a native speaker may not think that "Chloë" or "Chloé" is meant to be pronounced differently than "Chloe", and even if they do, no two speakers will necessarily agree on what the pronunciation is.
Culturally, people are entitled to dictate both the spelling and the pronunciation of their names. If your friend says that her name is pronounced "Clo-ee" and spelt "Chloé", then she is correct. But your other friend who says that her name is pronounced "Cluh" and spelt "Chloé" is also correct. And your other other friend who says that his name is pronounced "David" but spelt "Da5id" is correct too.
¹ as pointed out in the comments, names spelled according to the rules of another language are typically reproduced verbatim in English text, including diacritics if any, but English speakers generally won't know what the diacritics indicate unless they recognize and are familiar with the source language.