Renew
as it is used in this verse means,

"To make new; to renovate;
to transform; to change from natural enmity to the love of God and His law: to implant holy affections in the heart;
to regenerate."

Webster's 
1828    
I beseech you therefore, brethren,
by the mercies of God,
that you present your bodies a living sacrifice,
holy, acceptable unto God,
which is your reasonable service. 
And be not conformed to this world:
but be ye transformed
by the renewing of your mind,
that ye may prove what is that good,
and acceptable, and perfect,
will of God.

Romans   
12:1-2   

But how do we "renew" our minds?
Just like on This Old House, we need to renovate before the house is livable.
Remember how they tore down a wall here; added a bathroom there; ripped out a floor
or a ceiling and started anew?  Sometimes the whole foundation was rotten and needed
to be torn up and a whole new foundation put in.  In just this way, after we are Born Again,
we must renovate the "old house". To make our minds 'livable', we dig up the rotten foundation of our old life; tear down the old sinful thought patterns, rip out the old worldy habits, get rid of our unworkable, ungodly ways of dealing with life...  With Jesus as our new foundation, solid as a Rock, and the Holy Spirit leading the work, we can build the new
floors and ceilings of our minds with the solid material of Matthew, Mark, Luke,
and John; the new walls and rooms with the rest of God's Word, and then get started
on the gardens outside! 
This takes hard work, study, and time.  For the rest of
our lives we will never stop redecorating...a picture of EphesiansVI here,
repainting a wall with Jude there, adding more storage for
First and Seond Peter...
But, fortunately, God knew
that we'd find building better homes and
gardens to be wildly irresistable,
and something we, His children,
never seem to tire of!