Is there a theoretical minimum mass main sequence star that can become a red giant? Wikipedia's Red Giant page says they range in mass from 0.3 to 0.8 solar masses.   Which corresponds to an initial mass of a bit less than 1 solar mass, about 0.8 or so, if I was to estimate and a few solar masses.   Given that over half the mass is lost prior to the Red Giant stage.

Any stars with less than 0.8 solar mass to begin with wouldn't have had enough time to go red giant, so it's possible that significantly smaller stars could eventually go red giant.   Is there an estimate for how light the main sequence mass can be where the star still becomes a red giant late in it's life?
Perhaps you are interested in the lower mass limit (given in the comments as 0.3 solar masses), and why that lower limit exists.  This is similar to the question of why stars become red giants in the first place.  You are right that given the age of the universe stars less massive than 0.8 solar masses won't have had time, so the lower mass limit is theoretical.  But what sets it?

The answer is, red giants happen because stars can exhaust all the hydrogen in their core, turning it all into helium.  Then there is no nuclear fuel to keep the star in equilibrium, so no balance between the light that leaks out (which sets the luminosity) and the fusion rate (which mostly just responds to that luminosity).  As a result, the core shrinks and heats, and fusion initiates in a shell around the core-- a shell that used to be too cool to fuse.  

The temperature in that shell, and key aspects of its fusion rate, are determined by how small and massive the core becomes (the core gets smaller with time because it is losing heat, and it gets more massive because helium "ash" keeps being added to it from the shell).  Eventually the free electrons in the core become what is called "degenerate", meaning they approach their quantum mechanical ground state, and that implies the core gets about as small as the Earth. This also means the shell is extremely hot, and the fusion rate goes essentially berserk.  This requires the envelope to puff out into a red giant.

Hence, to have a red giant, you must have a degenerate core, and you must have shell fusion surrounding it.  But stars less massive than 0.3 solar masses have an internal structure that is purely convective.  Thus they don't run out of hydrogen in their cores until they run out of it everywhere  (and like you say, none have yet done this in our universe, it's a theoretical expectation).  That is why they cannot have shell fusion, and cannot become red giants.