The Oxford University Press collection of Sidney's major works has this to say about the title:
There is no evidence that the title is authorial. It derives from the first printed text, the unauthorized quarto edition published by Thomas Newman (1591). Newman may also have been responsible for the consistent practice in early printings of calling the lover persona 'Astrophel'. Ringler emended to 'Astrophil' on the grounds of etymological correctness, since the name is presumably based on the Greek aster philein, and means 'lover of a star' (with Stella meaning 'star'); the 'Phil' element alluding also, no doubt, to Sidney's Christian name.[5]
