I looked for a less mystical/legendary explanation, and found in Even Shoshan dicitonary, though without much detail, that /bin/ is a secondary construct form of /ben/. In modern Hebrew it survived in three similar set phrases: בן רגע, בן יום and בן לילה (within one moment / day / night). Because they are set phrases they are irregularly written without Yod also when writing without niqqud, but people know to pronounce it /bin/.
That is how the conjugation of pa'al works. There are three situations where the feminine form has ה in pa'al (if I am not forgetting something).
1) When also the masculine form has ה. For example - רוצה, שותה. They are written the same, but pronounced differently - rotze, rotza and shote, shota respectively.
2) The second situation is when it's one of those two letter verbs, for example בא, באה or שם, שמה - ba, ba'a and sam, sama.
3) When you have stative verbs, which have the pattern XaXeN instead of XoXeX in masculine singular, for example ישן, ישנה or גדל, גדלה - yashen, yeshena or gadel, gdela. There aren't too many of those, and the only way to know them is to learn them by heart.
The rest of the verbs follow the pattern XoXeX for masculine and XoXeXet for feminine - עומד, עומדת or לובש, לובשת - omed, omedet and lovesh, loveshet.