Well, it's not quite arbitrary. The fact that it falls around E2 is somewhat arbitrary, but the fact it falls on a note called "E" has a very long (and convoluted) history.
One relevant fact is that guitars (and their relatives like the lute and vihuela) rose to prominence in the renaissance. While a large variety of tunings were used for these instruments before they settled down around the late 1600s, there are still some important trends.
The first of which is that the standard renaissance scale had a low termination note of G, what we now notate as G2 (notated at the time with a Greek letter gamma, giving the note the name gamma-ut or gamut). Note that absolute pitch had not been standardized at all by that point -- A440 was still centuries away. So what was notated as "gamma-ut" might be a G2 or it might be a A2 according to modern pitch frequencies or it might be an E2 or anything around that range. But it was written on the bottom line of what we now think of as the bass clef, as a notated "G2." The point is that it was thought of the lowest note of the scale, hence why it is the lowest line on the lowest standard (bass) clef; notes below it originally were not "possible." As notated music at the time tended to be centered around vocal music, it was often around the lowest pitch for bass voices to reasonably sing. (The entire scale came to be known as the gamut after this lowest and starting note.)
The earliest treatise that gives any detailed description of guitars and their tunings is Juan Bermudo's treatise of 1555. It focuses more on the vihuela, but it gives insight into the state of guitars and general tunings for fretted instruments in Spain at that time.
Earlier in the renaissance, various tunings for fretted instruments were proposed and experimented with. Some tried to imitate the perfect fifth pattern of strings (like the modern violin family), while others tried patterns of perfect fourths. The latter tunings were more practical for playing chords. As the number of courses on instruments increased, a tuning in all perfect fourths could lead to the need to tune a string to a chromatic pitch (with a flat or sharp). Such notes were not viewed as part of the standard scale/gamut of the time, and were treated mostly as musica ficta, i.e., notes that weren't even notated consistently, but rather were expected to be chromatically altered or modified when needed for good musical taste. Tuning a fundamental string of an instrument to a musica ficta note was very unusual.
Also, for playing chords, it was apparently realized that inserting a third somewhere in the tuning was helpful. Hence, by the time we reach Juan Bermudo in 1555, the standard tunings he gives for the vihuela are similar to common lute tunings of the time, with a pattern of fourths and a major third in the center. The first tuning he provides begins on gamma-ut (and is labeled with a gamma to indicate it as the distinct lowest note of the scale), including the notes G-C-F-A-D-G. The second tuning is shifted up a whole step: A-D-G-B-E-A. The rationale for the latter tuning may have to do with the fact that it's one other place in the scale that would allow this pattern without accidentals, but it also likely has some correspondence to the fact that the ancient Greek scale (which the medieval scale was based on) terminated on a low A (proslambamenos) rather than Gamma.
In any case, those two tunings (G-C-F-A-D-G and A-D-G-B-E-A) had emerged as relatively standard options for six-course fretted instruments like the lute and vihuela in the 1500s. The former ultimately became a common tuning for lutes. Again, both tunings terminated at the lowest point of the standard scale of the time.
When the five-course guitar broke onto the European music scene in a huge way in the early 1600s, it often instead borrowed the vihuela tuning pattern from the second tuning, omitting the highest string, i.e., A-D-G-B-E. Note that tunings in the early 1600s on guitar frequently still jumped around and have overlapping strings within the same octave. (Some standard tunings as listed here.) I'm not sure that we know why the A-D-G-B-E tuning became more predominant on guitar, but it might have something to do with the reemergence of Greek theory and reinvention of ancient Greek music in the late 1500s, where the guitar played a prominent role. (Note that even the word guitar is etymologically derived from the Greek kithara.) Recall that A was the lowest note of the Greek scale, possibly emphasizing the choice for a tuning that terminated with that lowest note, hence A-D-G-B-E. The Greek music theory connection is also emphasized in the way that such tunings are generally listed in historical treatises, always tuning down from the highest string. Greek scales were similarly tuned, starting from the highest note and tuning a pattern of fourths below.
The final step was the development of the modern six-course guitar. While some such guitars imitated a lute tuning and effectively added a higher string (A-D-G-B-E-A), by the time the six-course guitar emerged, the old gamut with G as lowest note was no longer as relevant. Hence, the A at the bottom of the guitar was no longer thought of as a "lower limit" to the scale, but just an arbitrary note. It was therefore possible to just continue the pattern of fourths one more down to E, creating the modern standard tuning of E-A-D-G-B-E.
Bottom line is that there were good historical reasons why the second-lowest string on the standard guitar was named G or A, sounding as the notes at the bottom of the bass staff. The E on the lowest string was just added later to follow the pattern of fourths and expand the guitar's range.