Wikipedia tells us that Sri Lanka is Sanskrit for "sacred island". Their source - a New York Times article. Another page says it's Sanskrit for "resplendent land". Other sites and books repeat similar etymologies.
The Sanskrit श्री लङ्का śrī laṅkā means "splendid Lanka". I can't find any dictionaries that actually list laṅkā as meaning "island" or "land". In fact laṅkā was also the name of the capital city. As Apte says, it is the "N[ame] of the capital and residence of Rāvaṇa and identified with the island of Ceylon or the chief town in it".
However, the Tamil name for the country, இலங்கை ilaṅkai, also seems to mean "an inlet or island, an inlet formed in a river". So maybe the story that it means "island" has Dravidian origins. (But the general Tamil word for "island" is தீவு tīvu.)
The history of Sri is clearer. श्री śrī means "light, lustre, radiance, splendour, glory, beauty, grace, loveliness". It's also found in Srinagar (श्रीनगर "splendid city"). It's derived from the Proto-Indo-European root *ḱreiH- "to be outstanding, brilliant" (AHD). It is cognate with κρείων "noble, princely", and the personal name Κρέων - Creon, the king of Thebes in Greek mythology.
I pronounce sri with the same sound as at the beginning of shrine - with /ʃ/ instead of /s/ - just because it more closely approximates how I hear the word pronounced in Hindi, where it is commonly used as an honorific. But it turns out the pronunciation with /s/ is also standard, and is the one used by the CBC and BBC.
The logo for Shree Cement Limited is ingenious. It resembles both श्री and the English initials "SCL".