The earliest I've been up there was mid-to-late June and you could be anywhere from summery to full-out winter camping on 6' of snow. Your first issue may be getting to the trailhead if you plan on going up past Shadow Lake and Ediza Lake from Agnew Meadows. The Minaret Summit road sometimes opens by Memorial Day but often not until later.
Follow snow conditions at the
Mammoth Mountain ski area. So far this year, they've had about 434" of snow, which is higher than average for the past decade or so. Not up to the record (over 600") by any means but you'll probably be camping on snow above Ediza in May unless it really warms up fast.
For me, climbing "my" mountain on snow is easier than when it's mostly rock. Check out the '98 and '99 trip Mt. Ritter reports on my
web site for what I consider the "easy" way up from Ediza via the Southeast Glacier and the back chute up to the upper bowl. If it's all snow, it's a walk-up with ice axe and crampons. If the snow has melted out of the chute, it's still a straightforward scramble.
Good luck, and remember the First Rule of mountaineering: The summit is optional. Coming back in one piece is mandatory.