Firstly, it is important to note that the practice of having a lain rather than standing tombstone is found not just in Israel but historically across many Jewish communities in the Middle East / North Africa. That said, the question of what basis there is for it, if any, is an interesting one that some have attempted to address.
Rabbi Yoseph Schwartz in his Hadrath Qodesh (a work which in significant part deals with the rules and customs associated with the Hebhra Qadisha) records his correspondence with Rabbi Shelomo Aharon Wertheimer regarding the basis for the custom in Jerusalem/Israel to have a laying tombstone rather than an upright tombstone (as was traditionally done among most Ashkenazim in recent centuries). Rabbi Wertheimer similarly includes this correspondence in his work of responsa, She'eilath Shelomo vol. 2, no. 50. His record of the correspondence elaborates slightly more, so between the two I cite him below:
המצבות שעל הקברים בירושלם על הר הזתים יש על הקברים בבית הקברים הקדום
שלפני מאתים שנה במורד הר ציון הם מונחים על הקברים ולא עומדים כמו הנהוג
בחו"ל וכן הם המצבות הישנות שבשאר ערי אה"ק וטעם להשינוי הזה בלתי נודע
כי בכתוב מצאנו ויצב יעקב מצבה וכן מצבת קבורה רחל עד היום הזה ולשון
מצבה מורה על הנצב ועומד לא על המונח ושכוב כמ"ש ולא תקום לך מצבה ומצבה
לא תקימו ואולי עשו כן במכוון כדי שלא לחקות מעשה או"ה שהם מעמידים
המצבות בגובה או בעבור שבשכיבה מתקיימת המצבה יותר לימים רבים או שיש בזה
איזה טעם כמוס ע"ד סוד והנה על גולל ודופק פי רמב"ם ויקרא כסוי הקבר גולל
להיותו אצלו מאבן מתגלגלת משמע שהיו מניחים אז מצבת אבן על פי הקבר
בשכיבה לא בעמידה וכן בתוס בסנהדרין ט"ז ב מפרש ר"ת דגולל היינו אבן גדול
מלשון וגללו את האבן שמניחין על הקבר לסימן והיא מצבה וכן הוא בס הישר
לר"ה סי ס"ד וברש לאהלות פ"ב מ"ד ומלשונם זה שהוא על שם וגללו את האבן
משמע שמנהגם היה להניח האבן בשכיבה כמו האבן המתגלגל ומונח ע"פ הבאר
The tombstones that are on the graves in Jerusalem on Mt. Olives and
on the graves in the ancient cemetery from hundreds of years ago (on
the descent of Mt. Zion) they rest upon the graves and are not
standing as is practiced outside of the Land, and the same as regards
the ancient tombstones in the rest of the cities of the Holy Land. And
the reason for this change is unknown/perplexing, as we find “and
Jacob set up a מַצֵּבָה” (Gen. 35:14) and similarly the מצבה for
Rachel’s tomb which stands to this day, and the language of מצבה is
instructive of that which is perpendicular (נצב) and standing rather
than which is lain at rest, as is written “you shall not set up a מצבה” (Deut. 16:22) and “you shall not erect a מצבה for yourselves” (Lev. 26:1). And perhaps the
intent [of the custom to have a tombstone lain rather than standing] was to avoid imitating the gentiles who erect tall/high
tombstones, or perhaps because in a laying position the tombstone will
more likely endure for a long period, or perhaps it involves a closely
guarded secret on the basis of sod [i.e. esoterically/mystically
significant]. And we see that the Rambam as regards golel w’dofeq
(Ohaloth 2) explains golel as a cover to the grave made of stone
that is rolled over it and it is evident that they rested upon the
opening of the grave a rock tombstone that was lain upon it not
standing. And so too in Tosafoth Sanhedrin 47b cites Rabbenu Tam as
explaining that, based on “they rolled the stone” (Gen. 29:3) we see
that golel is a large stone which one places on the grave as a
marker, which is a tombstone. And similarly one finds this in Sefer
ha-Yashar of R”T 64, and the Ra”sh on Ohaloth 2 that this language [of
golel] is since they rolled/slid the stone and it is evident that they
were accustomed to resting the stone in a laying position just like
the stone that was laid upon and rolled/slid off the opening of the
well (Gen. 29).
Here we see, that R. Wertheimer did not have any received rationale for why the custom to have a laying rather than standing tombstone exists, nevertheless he furnished multiple plausible explanations:
- This was reactionary, i.e. it was consciously done in order to avoid imitating gentiles that erect large and looming monuments over graves.
- Practically a laying tombstone will weather time better, it is less likely to fall and break.
- There is some mystical significance to doing so that has been guardedly concealed.
- It is actually in perpetuation of ancient Jewish practice, as is evident from the Rambam and other mefarshim.
As for the third suggestion, that it is done on the basis of Sod (esoterica), the Mishkan ha-Nefesh (p. 141, fn. 13):
נראה שע"פ סוד מוכח שהמצבה היא שוכבת שהרי בצורת ההשתטחות על הקבר הורה
האר"י שצריך המשתטח לשכב ע"ג המצבה פנים אל פנים כנגד אבר וע"כ שהמצבה
שוכבת היא וכן המצבות בעירק בטורקיה הודו ומצרים שוכבות ולא עומדות וכן
אפשר לראות בביה"ק בארבל שהוא ביה"ק יותר מאלפיים שנה שישנם כמה קברים עם
מצבה שוכבת
It appears that according to Sod it has been demonstrated that a
tombstone ought be lain, since this is the form within which one
prostrates upon a grave as instructed by R. Isaac Luria. That one has
to prostrate on top of the tombstone, face to face, limb to limb, and
accordingly the tombstone is in a lying position. And such is the case
for the tombstones in Iraq, Turkey, India, and Egypt, where they are
lain and not standing upright. And it is also possible to see this in
the cemetery of Arbel which has a cemetery that is older than a 1,000
years old wherein there are tombstones that are lain.
Without getting into the nitty gritties of whether such a practice constitutes the biblically prohibited doresh el hamethim (inquiring of the dead) and whether this is something people ought be doing, this view maintains that having a tombstone that is lain rather than standing is intended to facilitate those that lay prostrate upon graves in order to commune with the soul of the interred. If you are interested in learning more details about this Lurianic practice, see R. Haim Vital's description in Sha'ar Ruah ha-Qodesh, p. 109. For multiple reasons, I do not personally find this particularly persuasive but the view is out there in circulation so it should at least be mentioned.
As for a non-textual/anecdotal source, if you are open to the possibility: Upright tombstones were typical of lands with snowy winters (and are thus more typical of the lands of Ashkenaz). I once heard it suggested that the reason for an upright tombstone is that it would remain visible and identifiable throughout most of the year, even when there was snowfall. In the Middle East / North Africa, where large snowfalls are more of a rarity the default remained to have the tombstone lain flat.