According to Adi Sankara's commentary on the Vishnu sahasranama, Sriman means One on whose chest Sri or Lakshmi, mother of the world, always dwells. Alternatively, it means One who is endowed with greatness of every kind or one who is more resplendent than everything.
When it is written as a name prefix (often in short as Sri), it is akin to Mr. in English.
1 The list of ten avatars varies regionally. The two substitutions involve Balarama, Krishna and Buddha is considered the avatar of Vishnu. Krishna is almost always included; in exceptions, he is considered the source of all avatars.