Can I have an incrementing count variable in LINQ?

Rather than using side-effects, use the overload of Select which takes an index:

stuff.Select((value, index) => new { index, value.Name });

You could do it using side-effects, but not in the way you tried:

int counter = 0;
var query = from a in stuff
            select new { count = counter++, a.Name };

I would strongly advise against this though.