What is the meaning and translation of my niece's name? My young niece, whose birthday is in principle coming up at the end of the month (Feb 29), is named 润润. When I look it up in Google Translate, it returns “wet”.  Other dictionaries show the same thing, even a guide that is specific to names.

Does this have some other meaning such as an idiom or context-specific usage, or is it a pretty sound, or what?  

Also, how is it properly pronounced? I hear a short vowel sometimes like the word “in”, but also seems to sound very close to “run” more often.  Given that English is tone deaf, what at least is the best American English sound to use?

(They are in Kunming, Yunan. I don't know the differences between college Mandarin and local Yunaneese, but that's another question.)
Probably it is because it was a leap year (闰年). 闰 and 润 have same pronunciation, which is similar to English word 'ruin' (even the tone is the same) but shorter.
闰 means intercalation and 润 means moist, smooth literally and nurturing, nourishing/nourished prosperity and affluence figuratively. 润 is often a good word and commonly used in people and organization names.