The -mir ending of Faramir's name is almost certainly 'jewel' or 'precious thing', but Fara- is much more difficult to translate. The Elvish root far- means 'sufficient' or 'adequate', so it may be that the brothers Boromir and Faramir have names related to their father's attitude toward them. As Denethor's favourite son, Boromir was perhaps the 'faithful jewel', while the less favoured Faramir was merely the 'sufficient jewel'. These speculations, of course, belong to the realm of guesswork, since Tolkien makes no definitive statement about the names' origins.
Though it's established beyond doubt that Faramir and Éowyn had a grandson named Barahir, their son - and Faramir's heir as Prince of Ithilien - is never conclusively identified. In the drafts of the Appendices to The Lord of the Rings, given in volume 12 of The History of Middle-earth, he is shown as 'Elboron'. However, no name is given anywhere in the canonical works, and it is difficult to be sure whether Tolkien meant the name 'Elboron' to stand or not.