The date of Orodreth's birth appears only in The History of Middle-earth volume 12, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
Steward Orodreth was doubtless named for the famous Elf-lord Orodreth of the First Age, who ruled in Nargothrond after the loss of his brother Finrod. The detailed etymology of the name is unclear, but it is known to contain the word orod, 'mountain'.
Orodreth's sister Morwen is only named in volume 12 of The History of Middle-earth. According to that source, when Orodreth's son Ecthelion died without issue, it was Morwen's descendants, and specifically her grandson Egalmoth, who continued the line of the Stewards.