Having run the Mother Road in a big truck when much of it was still an official highway I guess I'll chime in.
I'd do it in the late spring. Lots of greenery... weather can still be iffy in the higher elevations until late May...I've seen snow in Flagstaff in July!
Take plenty of time... my bride and I ran from Bloomington, IL to Tulsa several years ago and it took us three days...Chicago to the Santa Monica pier can't be done properly in less than two weeks... too much to see.
There are lots of chunks in IL that run along I-55... some of the interstate was 66 around there. many antiques shops, cafes and such. Lots of dead ends everywhere... get a good map. We did u-turns in many folks driveways when following the earliest (1926) routes...
Above all have fun... it's a helluva ride... and don't miss Oatman Pass or the Chain of Rocks Bridge... or The Blue Whale... or.....