In the entire Tanach the name Pinchas פינחס is spelled with a Yud. The only place the Yud is missing is שמואל א: א. ג.
ועלה האיש ההוא מעירו מימים ימימה להשתחות ולזבח ליהוה צבאות בשלה ושם שני בני עלי חפני ו*פנחס* כהנים ליהוה
There are 2 other Pesukim (שמואל א: ד. ד. & שמואל א: ד. יא) where the Bnei Aili are mentioned by name and there the spelling is also with a Yud.
Why in this one location .שמואל א: א. ג is the Yud left out?
The Minchas Shai that msh210 referenced in his comment in turn refers to Paaneach Raza, who explains as follows (paraphrasing a lot):
The letter yud symbolizes the holiness conferred on a Jew by circumcision and the associated weakening of carnal pleasure (compare Rambam, Moreh Nevuchim 3:49). Appropriately, then, Pinchas, who risked his life to punish Zimri's transgression of this area of Jewish sanctity, deserves to have a yud in his name.
By contrast, Chofni and Pinchas, Eli's sons, are described as "wicked people" who "lay with the women who gathered at the entrance of the Tent of Meeting" (I Sam. 2:12,22). R' Yonasan, in the Gemara (Shabbos 55b), explains that this isn't meant literally - their sin was that they "delayed bringing the bird-offerings" [offered by women after childbirth or unusual discharges], forcing these women to wait in Shiloh until their offerings were brought - but that in G-d's eyes this is tantamount to adultery, in that it was an interference with these women's marital life. The Gemara there continues by citing Rav's opinion, that Pinchas definitely can't have been guilty of actual adultery, since we find distinguished descendants of his serving as kohanim (compare Malachi 2:12); but that Chofni may well have committed such a sin, and Pinchas is held liable too, because he could have protested (and gotten Chofni to stop this behavior) but failed to do so.
Paaneach Raza goes on to explain that this issue is reflected in the variance between the Eastern and Western mesoros as to the spelling of פ(י)נחס in I Sam. 1:3. (Though note that Minchas Shai says that there is in fact no such variation.) The letter yud also represents yad, a hand (such as might be used to stop someone else from sinning). Thus, he says, this variation can be explained in any of the following ways:
The Western mesorah omits the yud from Pinchas' name there, to symbolize the fact that he failed to use his yad to stop his brother Chofni's actions. The Easterners keep it there too, to show the same thing but in reverse: he had a yad but failed to use it.
The Westerners perhaps hold that Pinchas himself was guilty of some kind of immoral behavior (he doesn't say whether to the extent of the surface meaning of the verse, or just the exposition above about the women's offerings), and therefore lost his yud. The Eastern tradition holds with the Gemara, that Pinchas himself was blameless, and is taken to task only for not protesting his brother's actions.
R' Yonasan would say (with the Easterners) that there's no reason to omit the yud from Pinchas' name, since after all his sin was only the delay in offering the sacrifices. While the Westerners, who omit the yud, would hold with Rav that Chofni was indeed guilty of actual adultery, and then Pinchas is held to blame for not acting like his namesake and stopping it.
I like Alex's answer.
I would suggest that, according to the Zohar, it is quite possible that only one Pinchas (the one in Parshat Pinchas with the small yud) in the Torah is written plene, that is, malei yud. And all others it is written deficient, that is, chaser. See my two posts (here and here) on this matter, basically presenting Minchas Shai on Pinechas, where he strives to harmonize our present texts and the Zohar.
If so, the difference between Torah and Nach is that Torah is earlier, and in general uses more chaser spellings, while Neviim and Ketuvim were written by different authors and later, when consonantal vowels were more in vogue. I have no ready explanation, within Nach, why there is one exception, just as no ready explanation why, in Torah, there is one exception.
Well, I do have one. I don't know that you would like the speculation. But small yuds are to be as if not read (see e.g. the small aleph in Vayikra, and I can offer other examples as well). There might be the one exception in Shmuel to indicate its classic spelling, before diverging into the comfortable. And a parallel in the famous case in parshas Pinchas. Or something along those lines.
Update: To clarify, I was just answering according to Zohar's masorah, not according to the one established by Minchas Shai. Alex answers according to the one established by Minchas Shai, though I will add a bit below now.
Also, As Rav Yosef says in Kiddushin 30a, א"ל: אינהו בקיאי בחסירות ויתרות, אנן לא בקיאינן, that we are not expert in plene and deficient spellings of words. As such, I would be wary of making big deductions out of malei / chaser spellings across Tanach which might not even be accurate, and which Chazal themselves admitted as much.
But, assuming it is as established by Minchas Shai, a counterweight to a midrashic / kabbalistic explanation, as provided by Rabbi Avraham Ibn Ezra, in commenting on the divergences in the two accounts of the aseres hadibros:
וכאלה רבים ושניהם נכונים, כי הכתוב בלא וי"ו אחז דרך קצרה ולא יזיק. גם הכתוב בוי"ו לא יזיק בעבור שהוסיף לבאר. והנה הוי"ו שהיא נראית במבטא הפה אין אדם מבקש לו טעם למה נגרע ולמה נכתב ולמה נוסף. כי זה וזה נכון. והנה על הנראה שיבוטא בו לא יבקש עליו בכתב טעם. א"כ למה נבקש טעם בנח הנעלם שלא יבוטא בו. כמו מלת לעולם למה נכתב מלא או למה חסר. והנה בני הדור יבקשו טעם למלא גם לחסר. ואילו היו מבקשים טעם לאחד מהם. או שהמנהג היה לכתוב הכל על דרך א' הייתי מחריש.
והנה אתן לך משל: אמר לי אדם אחד כתוב לרעי וזה כתוב: אני פלוני אוהבך לעולם. וכתבתי פלני בלא וי"ו. אהבך גם כן בלא וי"ו. לעלם חסר. ובא ראובן ושאלני למה כתבת חסרים?! ואני אין לי צורך לכתוב רק מה שאמר לי. ואין לי חפץ להיותם מלאים או חסרים אולי יבא לוי ויודיעני איך אכתוב. ולא ארצה להאריך רק המשכיל יבין ועתה אפרש לך השאלות הנזכרות.
"And there are many like these, and both of them are correct. For the Scripture without the vav chose the short way, and does no damage. Also the Scripture with the vav does no damage, since it added so as to explain. And behold, the vav, which appears in the pronunciation of the mouth, no man seeks for himself a reason why it is missing and why it is written, and why it is added. For both this and that are correct. And behold, it is apparent that the one who pronounces it would not seek upon it, when written, a reason. If so, why should we seek a reason for the omission of that which is not even pronounced. Such as, in the word le'olam, why it is written plene (with a vav) or deficient (without)? And if we were to seek a reason for only one of them (malei or chaser), or if the general custom was to write all of them in one way (malei or chaser), I would have remained silent.
And behold, I will give you an allegory. A person says to you, "Write to my friend, and this is what you should write: I, Ploni love you forever." And he writes Ploni, פלני, without the vav {which is acceptable in Hebrew}, and אהבך, 'I love you', also without the vav. And לעלם, "forever" also chaser. {In each case you can have a cholam chaser as opposed to a cholam malei.} And then Reuven comes and asks me, "why did you write them deficiently {chaser}? And meanwhile, I only have need to write that which he said to me, and I have no desire that they be either malei or chaser. Perhaps Levi will come and inform me how I shall write. I do not wish to go on at length. Rather, the intelligent person will understand."
