If one took down mezuzas for renovations and the doors where they originally were have been moved does one make a new bracha when putting them back up. What is the source?
Yes, a new beracha is made upon affixing the mezuzos again.
The principles for this question are derived from other halachos – those of Tallis and those of Mezuza.
Concerning Tallis we find the Rema ruling that if one removes a Tallis with intent of putting it back on, a new beracha is not recited, unless a significant hefsek is made.
For purposes of mezuzah the Aruch HaShulchan mentions removing a mezuzah and replacing it the next day (for purposes of checking it), in which case a beracha is made; the Shulchan Aruch HaRav (8:23) mentions “several hours” for purposes of Tallis and for Tefillin notes “two or three hours” (25:29).
In terms of renovations, there will usually be a long period of time, and a new beracha is therefore recited.