F.ex. there would be Yitzchak, but sometimes it would be Yitzchak Ajzik.
Why did the gravestone include a nickname?
This situation reminded me of a conversation I once had with a rabbi who asked me in detail what my parents called me, what my friends called me, what was written in my identity papers.
It is true that in Israel some still have the name Yitzchak in their identity papers and are called Ajzik by their family and friends. I had just thought of it as a cute Yiddish-sounding nickname.
But Jewish life in Diaspora has other challenges than living in Israel, both in the past and today, regarding given names. F.ex. in the States today many Jewish families will choose a modern American sounding name for their child, and give a second traditional name commemorating a deceased ancestor, if they are Ashkenazi. Sometimes the child will be called by an additional third name, a nickname, as a sign of affection.
Imagine similiar situations facing the Jews living in prewar Lubaczow.
The baby boy would get a Hebrew given name at his circumcision with his father's Hebrew given name in addition.
F.ex. Yaakob Yehuda ben (son of) Shmuel.
יעקב יהודה בן שמואל
It may be that his family and friends affectionately would call him Yankele / Jankele.
It may be that when he went to school he would write his name as Jakob or Jakub.
It may also be he started to use a completely different given name - a modern name in Polish or in German, in his grownup professional life, in particular if he moved to another place, to another country.
But in his Jewish life - at his Bar Mitzvah, in the synagogue, when he got married - he would be known as Yaakov Yehuda ben Shmuel.
יעקב יהודה בן שמואל
When he died, sometimes it seems the family felt it important not only to write down the Shem HaKodesh (the Sacred Name) but also the nickname on the gravestone. The nickname is called Kinnui in Hebrew.
A few times I have seen the combination of Avraham Avish / Abish on some gravestones. The second name has reminded me of the Hebrew name Avishai, but according to Alexander Beider Abish / Avish is the Kinnui for Aba (Abe) and was sometimes considered a Kinnui for Avraham because of its phonetic and semantic closeness to the name.
Would that mean that in this case this man originally had been named Avraham as a Shem HaKodesh and then later had the nickname Abish/ Avish?
To my understanding, if his sacred names were Avraham Aba, and he was nicknamed Abish / Avish, Aba would also have been noted on the gravestone.
In other words, I lean to the explanation that Abish in this case was the Kinnui for Avraham.