The word "shaf'" literally means adding something to something else.  It also means "even" as opposed to odd. The Quranic verse says "wash-shaf'a wal-watr" in which "shaf' means even while "watr" means "one".[1]  " Shafa'ah" (intercession) is derived from "shaf'" which means adding something to something else.[2]
The reason this prayer is called "shaf'" is because it consists of two rak'ats.  Some exegetes have said in the interpretation of the following verse:
«مَنْ یَشْفَعْ شَفاعَةً حَسَنَةً یَکُنْ لَهُ نَصِیبٌ مِنْها وَ مَنْ یَشْفَعْ شَفاعَةً سَیِّئَةً یَکُنْ لَهُ کِفْلٌ مِنْها وَ کانَ اللَّهُ عَلى‏ کُلِّ شَیْ‏ءٍ مُقِیتاً»[3]
" Shafa'ah" (intercession) is derived from "shaf'" which means "even" as opposed to odd - the interceder adds his own recommendation to the plea of the petitioner; in this way the number of pleaders becomes even, and the weak plea of the petitioner is strengthened by the prestige of the intercessor.[4]
As for the last part of the question, we must say: Well-being and good health are the fruits of the night prayer as stated in some traditions. For further information about them, you are advised to study the following indexes:
"The Importance and Effects of the Night Prayer and the Rules regarding Its Qaza", question 60928.
"Night Prayer", question 60929

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