How to pronounce ë in a name?

The mark on the letter "ë" and other vowels like it can actually be one of two things:

• A mark of vowel-fronting (often called an "umlaut," which is the term for the process). This is what it means in German but isn't really used that way in English.
• A diaeresis, which denotes that the vowel marked does not form a diphthong with the vowel before ("hiatus").

In English, it is normally the latter:

The grave accent [`] and the diaeresis are the only diacritics native to Modern English (apart from diacritics used in loanwords, such as the acute accent, the cedilla, or the tilde). The use of both, however, is considered to be largely archaic. (Wikipedia)

Furthermore,

The diaeresis mark is sometimes used in English personal first and last names to indicate that two adjacent vowels should be pronounced separately, rather than as a diphthong. Examples include the given names Chloë and Zoë, which otherwise might be pronounced with a silent e.

Here I'll add some IPA

For example, "Chloë" /kləʊ.i/ and "Zoë" /zəʊ.i/ do not rhyme with "toe" /təʊ/, even when written without a diacritic. They have two syllables (indicated by the . break in the IPA, which is a casual notation) compared to one.

In American English (and I suspect other dialects, but I'm hesitant to say absolutely), they rhyme with "doughy" /ˈdoʊ.i/, as in:

That roll was awfully doughy.

As the previous post indicates, it is used to indicate that the vowel is pronounced in hiatus, not as a diphthong.

It is also sometimes used to romanizing words from languages (Russian, Belarusian, Mongolian, etc.) written in Cyrillic: Cyrillic ё indicates the "yo" sound [jɵ]. For instance, it should really be Gorbachëv (it's [jɵv] not [ev]), and Artёm or Artyom are both possible transliterations of the male given name Артём.