Etymology can be a knotty field! I've been sort of mentally keeping track of muddled associations I've spotted on forums and baby-name sites, and finally decided to unravel them all (mostly for myself, but I hope others might enjoy, too). :)
Amelia and Emilia--Amelia is Latinized Germanic, meaning "work, vigor", while Emilia is from Latin, "competitor, rival". They may be pronounced nearly identically in English, but in many languages, they are quite distinct.
Cecelia and Celia--both are Latin, but Cecelia is from Caecilia ("blind") and Celia from Caelia ("heaven").
Eloise and Louise--both are French forms of Germanic names, but Eloise ultimately comes from Helewidis ("healthy and wide" or "healthy wood") and Louise from Chlodovech ("famous battle").
Ellen/Elaine/Helena and Eleanor--Helena is from Ancient Greek, prob. "torch". Ellen and Elaine are medieval English and French forms, respectively. Eleanor comes from Provençal Alianor, itself probably from a Germanic name, like Aldenordis ("old north").
Genevieve and Guinevere--both are medieval and Gallicized, but Genevieve is from Germanic Genovefa, prob. "tribe woman", while Guinevere is from Welsh Gwenhwyfar, prob. "white phantom".
Jerome and Jeremy--Jerome is Anglicized from Greek Hieronymos ("sacred name"). Jeremy is a medieval form of Hebrew Jeremiah ("Yahweh lifts up").
John and Jonathan--well, they're not completely unrelated: both are Hebrew and theophoric, but John didn't originate as nickname for Jonathan. John comes from Yehochanan ("Yahweh is gracious"), and Jonathan from Yehonatan ("Yahweh has given"). For some reason, John went through a lot more changes passing through Greek and Latin than Jonathan did!
Livia and Olivia--Olivia is a Shakespearean coinage, probably based either on Oliver or Olive (which are, incidentally, also unrelated to each other, see below), while Livia is a much older Latin name, poss. meaning "bluish, jealous".
Olive and Oliver--shockingly, Olive means "olive", taken almost directly from Latin ;) Oliver is Anglicized from French Olivier, itself from Old Norse Olaf/Áleifr ("ancestor's descendent") or from a Germanic name like Alfher ("elf warrior") or Aylward/Adalward ("noble defender").
Rose and Rosalind/Rosamund--although they're commonly interpreted as "beautiful rose" and "rose of the world", respectively, in Latin, Rosalind and Rosamund are not Latin names, but Germanic. The 'rosa-' part is actually derived from hros, "horse" (and there were many other Germanic -hros- names, but these two are by far the most well-known today): Rosalind is from Roslindis ("gentle horse"); and Rosamund from Rosmunda ("horse protection").