a_p - yeah, I know there's more to Switzerland. OP could select for example the Valais region or maybe check out the Aletsch glacier in the, er, Aletsch region - neither is too far from the Bernese Oberland as the high altitude capable crow flies.
The reason I pick the Bernese Oberland all the time is because most people tend to pick this as their mountainous region the first time they come (Swiss tourism stats backed that up about 10 years ago when I looked, I assume it's still the case). Also, public transport is pretty good for this region - it's not so hot in the Aletsch glacier. Assuming the OP is flying from somewhere in N America, they'll probably route into Zurich or Geneva and it's relatively quick and very easy to get a train to Interlaken East or West station and change for local onward travel which is well established. There's also a fair few options for alternative activities if they want a day off hiking.
As for the hiking, end of May into June I'd expect the snow line in Switzerland to be around the 2000 metre mark (random source example: http://las.physik.uni-oldenburg.de/eProceedings/vol01_1/01_1_kleindienst1.pdf). Given that Grindelwald village itself is at 1,050 metres they'd have to climb a fair bit to hit the snow line. There's 300km of trails around that village so they can easily choose/avoid this scenario if they wish, it gives a lot of options to them.
I'll admit I do know the Bernese Oberland well so am biased. Sorry about that. I'm open to alternative suggestions, please feel free. :/