Having read more than a few books on medieval England I have come across the phrase "mysterious illness" dozens of times, but no further explanation, or even a description is ever given.
As I understand it he and Catherine of Aragon wed in 1501, then went to the Marches to establish their household. Shortly after, both fell ill with the same "mysterious illness". Catherine recovered, but Arthur did not.
Are there contemporary sources that describe the illness in general? Are there any modern sources that make educated guesses on its nature?
According to The Receyt of the Lady Katherine:
[Arthur suffered from] the most pitiful disease and sickness that with so sore and great violence had battled and driven in the singular parts of him inward; that cruel and fervent enemy of nature, the deadly corruption, did utterly vanquish and overcome the pure and friendful blood, without all manner of physical help and remedy.
Frustratingly there's a real dearth of information about the general health of Princes Arthur and Henry (one theory is that Henry VII was hyper-aware of how vulnerable his grip on power was to challengers in the aftermath of the Wars of the Roses, so kept illnesses on the DL as a potential sign of weakness) which is annoying for us because it makes modern diagnosis very very difficult.
Compared to the wealth of articles on Henry VIII's grotty leg or Katherine of Aragon's fertility in medical journals, this earlier period is woefully under-documented and therefor woefully under-diagnosed.
For a historian's perspective, then, the contenders are:
That most Tudor of lurgies: the sweating sickness, or "English sweate." But sweating sickness usually came fast and hard, whereas Prince Arthur was ill for weeks. Furthermore Robert Hutchinson argues in Young Henry: The Rise of Henry VIII that Arthur's death in 1502 is inconsistent with the five known outbreaks of the disease - unlike Anne Boleyn's attack which occurred in 1528.
Hutchinson posits possible causes as "an undefined plague - almost certainly bubonic" which was raging as little as 36 miles away from the Prince's household at Ludlow Castle. Plague is backed up by both contemporary Spanish chronicler Andres Bernaldez and reports that Henry VII didn't attend his son's funeral... in the cathedral of plague-riddled Worcester.
Hutchinson's preferred killer is tuberculosis/consumption, which could explain his inability to consumate the marriage to Catherine. David Starkey is also on board with this, but he also reckons this could be evidence of...
Testicular cancer, but this doesn't fit with the simultaneous illness of Catherine of Aragon and other members of the household, nor the description of "a malign vapour which proceeded from the air" which suggests a real belief in something contagious.