A Hebrew name is chosen, generally by parents, as are other names. A convert to Judaism chooses a Hebrew name, too.
Jews have a tradition of naming after someone special who has gone before, as a way of perpetuating that person’s “good name.” Ashkenazi Jews generally do not name after living persons. (Yes, there are examples of those who have done so.)
There is no “last name” to a Hebrew name. A person is XXXXX son/daughter of YYYYY and ZZZZZ. Thus, when one has a baby, one gives a first and maybe a second name - my mother called me “Hannah Nekha” - and the parents’ names follow thereafter: “Hannah Nekha bat Yehudah veRaizabella,” a mouthful. Note that my first name and my dad’s first name are Hebrew, and my second and my mom’s are Yiddish. We include the Yiddish as “Hebrew names.”
The Hebrew name may or may not be related to the English name. For instance, our friend Naomi’s English name is Naomi Lynn Smith. Her Hebrew name is Naomi Leah bat Hayim veHannah: Naomi carries through, but not Leah. To use my name again, my English name is Audrey, which was after my grandfather Adolph, regardless my Hebrew name.
If one is naming a baby (or oneself) after someone, it is often the practice to spell the name in Hebrew the same way the predecessor spelled it. There are quite a few Hebrew names (usually of Yiddish derivation) that have varied spellings. Thus when the baby is named, one should be certain to write the name somewhere and keep it in a safe place.
Synagogues do keep records (and we would be glad to keep your Hebrew names in our database), but because files for baby namings and weddings and such often stay with the rabbi who officiated, they may no longer be available to the synagogue. Of course, many times the synagogue is long out of business. And even if the shul has wedding records, there may be no copy of the ketubah (the marriage contract), because it may have been created by an outside craftsperson. (One might look to wedding photos to enlarge a shot of the ketubah, as film photos have molecular-level resolution.)
Many times the gravestone will have the Hebrew name, but definitely not always. Sometimes folks did not know the Hebrew name when the stone was made, and other times it has been family tradition to use only English.
Thus when folks call the office asking, “My name is Louis, so what is my Hebrew name?” we have only disappointing news. When they state, “My mother was a member there in the 1940s and she just died, so we need her Hebrew name,” we can offer only condolences. And when they ask us to look up their own naming in 1932, they are dismayed that the records may be hard to find.
Please collect your family’s full Hebrew names and keep them somewhere safe, written in Hebrew to the best of your research. (We can help with standard names.) And please make certain your children and siblings have copies.