The highly recommended suggestions by Stephan and aschipfl applied to your code:
@echo off
set /P "dest=Enter name: "
set "findest=Z:\ProjectIT\copy\%dest%"
%SystemRoot%\System32\robocopy.exe Z:\ProjectIT\copy\xcopy "%findest%"
See for example answer on Why is no string output with 'echo %var%' after using 'set var = text' on command line? for an explanation why set "VAR=Value"
on assignment and "%VAR%"
on expansion/reference is highly recommended even if the string to assign to a variable contains 1 more quotes.
Here is also an improved version which allows the string to be entered with double quotes and with each /
(Linux/Mac directory separator) replaced by \
(Windows directory separator) and which removes last backslash from destination path because of ROBOCOPY interprets a single backslash at end of a directory path as escape character for the double quote:
@echo off
setlocal EnableExtensions DisableDelayedExpansion
:EnterName
set "dest=""
set /P "dest=Enter name: "
set "dest=%dest:"=%"
if not defined dest cls & goto EnterName
set "dest=%dest:/=\%"
if "%dest:~-1%" == "\" set "dest=%dest:~0,-1%"
if not defined dest cls & goto EnterName
set "findest=Z:\ProjectIT\copy\%dest%"
%SystemRoot%\System32\robocopy.exe Z:\ProjectIT\copy\xcopy "%findest%"
endlocal