Ar-Rafi is the One Who may exalt anyone by His favours, gifts, blessings and grace.
To try to emulate the name you should try to extol and exalt truth and love those engaged in the pursuit of the truth and righteousness.
Appropriate recitation can provide protection from enemies and can be conducive to material prosperity.
see 'The 99 Most Beautiful Names of Allah' by Dr Zahurul Hassan Sharib
Some other references:
...whoever elevates his vision above the tangible and imagined things, and his intention above blameworthy desires, God has raised him to the horizon of the angels close to Him,..
(Al-Ghazali - The Ninety-nine Beautiful Names of God. Trans. Burrel/Daher: ITS: 1992.)
The elevation of rank can relate to the sensory and phenomenal world or to the unseen world. The advancing sufi gives up concern for elevation in this phenomenal world in favour of elevation in the unseen world - the realised sufi is able to put both into their proper perspective.
Bismillah ir Rehman ir Rahim
The Exalter The One Who Exalts Anyone, The Elevator