Hi Shilpa -
The Jungfraujoch is open all year and there will surely be snow there in May. The cograil runs all year, every day.
This map shows the cograil train line and various kinds of cableways:
map.search.ch/?poi=bergbahn,haltestel le,zug&pos=639040,1594 80&z=32&poi_id=404ac4e d58e523b95822-bus
You can turn of the transport overlays in the menu under Points of Interest.
The higher altitude trails will generally be blocked by snow, but the snow in general will not be of good quality for skiing. So, some hotels and tourist services will be closed from sometime in late March or April until mid to late June. They will open at various times, or, sometimes in weekends, some starting in May. Some hotels do not close for "low" season.
For example - the cableway from Wengen to Männlichen will be closed in early May, but open starting on May 20th. Use the timetable and check various dates. That particular service runs from Wengen Wengiboden up to Männlichen LWM station. I think your son would enjoy that...see attached images.
Usually, the very nice trail from Männlichen to Kleine Scheidegg is groomed for hiking when the snow is heavy in the Winter months, but not as the weather warms up, and then it tends to be cleared for walking starting in early to mid-June.
Other trails at lower altitudes will be open -- almost surely on the valley floor near Lauterbrunnen.:
A particular hotel that I like in Wengen will only open on June 8th.
My grandson and I were up there at Männlichen on June 15 in 2015. The trail was open, but there was lots of snow still left on the ground. The trail was open or closed on different days at the beginning of the season, because the snow could slide down the hillsides and block the trail in some places. And, at that time, there were thick clouds which only opened up for very short intervals.
The clouds are the issue ...if you only go for a short stay in the late May to early June period, less so as July and August come along, the clouds may block all views for a day or so before they clear out. You will still be able to find snow. You may not see it falling. The weather can change very quickly...good in the AM and cloudy in the PM, or vice versa. Weather forecasts are not reliable more than day or so in advance in the high Alps. Plan on being in the area for more than one day to improve your chances for good visibility.
If you stay at Interlaken, you have good train connections to the rest of Switzerland as well as to the Jungfrau. Interlaken is not much of a tourist destination in its own right, but it is a good transportation hub.