An augmented sign can be '+', which signifies a larger interval in an augmented chord in comparison to , say, the major. But where did the 'o' sign originate for the opposite - diminished? Is there a relevance to it?
Unfortunately, there's no clear indication why the degree symbol (°) was chosen to represent diminished harmonies. We do have some information about when and how it originated, though; specifically, it arose in conjunction with the use of Roman numerals in harmonic analysis.
During the Baroque period, harmonies were notated in shorthand using figured bass. Not until the publishing of Jean-Philippe Rameau's "Treatise on Harmony" in 1722 did musicians begin to notate harmonies according to their scale degree rather than the actual root pitch. While Rameau used phrases like "note tonique", "seconde note du ton", "mediante", etc. to name each scale degree, the length of the names (especially when used repeatedly) prompted the use of various shorthands instead. By the start of the 19th century, a variety of such systems were in use, as David Damschroder describes in his book "Thinking about Harmony: Historical Perspectives on Analysis" (page 7):
Roman-numeral, Arabic-numeral, and non-numeral strategies competed as notation for harmonic analysis. ... The Roman numerals in Gottfried Weber's Versuch einer geordneten Theorie der Tonse[t]zkunst (1817-21) appear in one of three guises - capital, small, or small preceded by a degree circle - to indicate each triad's quality (major, minor, or diminished, respectively).
This usage of the degree symbol by Weber to denote a diminished chord was, if not the earliest such usage, certainly pivotal in popularizing and standardizing it. As Mark Ellis describes in "A Chord in Time: The Evolution of the Augmented Sixth from Monteverdi to Mahler" (page 155):
By training, Weber was a lawyer and his work reflects a mind set on defining and classifying. For example, he states that there are seven basic chordal sonorities: major, minor, and diminished triads, and then these three triads each with a minor seventh, and the major triad with a major seventh. In addition, Weber was one of the first to develop an extensive vocabulary of symbols for describing the nature of each type of chord; he also popularized the use of Roman numerals to identify chords.
Sadly, in his text Weber leaves no clues as to why he chose this particular symbol to denote a diminished triad, simply stating:
3.) Den verminderten Dreiklang zu bezeichnen bedienen wir uns eben solcher kleinen Buchstaben, und hängen noch ein kleines Null oben an...
Or in English (tr. James F. Warner, 1851):
(3.) To denote the diminished three-fold chord, we will use the same small letters with a little cypher prefixed...
Thus, we can only speculate at the motivation behind choosing this particular symbol over any other. It certainly has benefits in its simplicity (both for printing and writing). But perhaps most importantly, it has no other musical meaning, and so avoids confusion - which is a key objective of any symbolic notation.