The DEFLECTION of your mandrel is quite independent of the type of STEEL it is made of.
However, heat treated steels can take larger deflections before the deformation becomes permanent.
My choice of steel would be SAE 4140 heat treated to Rc 44 to 46, then finish ground. If that is not possible heat treat it to Rc 38 t0 40 and finish turn using tungsten carbide tooling.
If all else fails make the mandrels from SAE 1144, also known as "stressproof". Not as good as the options above but far better than using cold drawn steel rod or soft (ie. non-heat treated) drill rod.
Just to re-iterate: This will not reduce the deflection but will reduce the likelihood of permanent deformation.
To reduce deflection (vibration) the mandrel would have to be made larger in dia., or shorter in length, or made from a "stiffer" material such as tungsten carbide. Unfortunately these are the only options available for increasing stiffness. But as stated earlier, using heat treated steel reduces the chances of permanently bending, and scrapping, the mandrel.
Trust this helps.