Stephen Lark kindly referred me to the Hull geneology site which suggests that Sir John was the legitimate brother of the 4th earl Roger (killed 1398) and Sir Edmund. However, further digging about on the web reveals that the 3rd earl only referred to two sons (Roger and Edmund) in his will and therefore the Hull site has almost certainly got to be wrong.
Elsewhere on the web Douglas Richardson points out that if John Mortimer was male heir to the Earl of March (as he allegedly claimed) he would have to be of legitimate descent. This is true as far as it goes, although people making wild claims rarely bother about technicalities. After the 5th Earl of March's death, his claim to the throne went (through his deceased sister, Anne) to his nephew, Richard, Duke of York. The Salic law cannot be used here, as that would make the Lancastrians legitimate kings and wipe out the Mortimer claim altogether! I suppose that an extreme stretch he might have had a claim to the title, Earl of March, but the lands were evidently not entailed to the male heir so this would not have helped him much. But if we assume Sir John Mortimer was of legitimate descent, the most likely explanation is that he was a grandson of one of the two younger brothers of the 3rd earl. Unless you know better?
But, hang about. It was the 3rd earl who married Philippa of Clarence. So his younger brothers and their heirs might have some sort of claim to the Mortimer lands and titles but they would have no claim whatever to the throne as they were not descended from Lionel of Clarence. I still can't make sense of this - at the moment I can't see any way that Sir John Mortimer had any right to the throne, but if he didn't why did the Lancastrian government consider him such a threat?