It is a name which originated in Southern India. maNi, which is cognate with English mane, means an ornament, a jewel.
venkaTa (that N is actually nasalized velar as in English -ng- sound) is the name of a mountain in South India, seems to have been alternatively written as vyankaTa. I do not know the etymology, but venka has been described as a tribe from South India, and the ultimate origin may be non-Indo-european (I know too little of any Dravidian or Austrasiatic language to hazard a guess). In any case, viSNu, the protector amongst the Hindu trinity, was identified with the deity that lived in these mountains (also, variously, shrIpati, tripati or tiupati). Sanskrit Isha means master, from Ish, to possess, so, venkatesha is another of his names.
The -n ultimately derives from a nominative singular masculine case ending.