Nepal's monsoon typically starts in June and ends in early September. September isn't as wet as July or August, but rain should still be expected from time to time. By the middle of the month, it's usually only raining every few days rather than daily, and by the end, it should almost have stopped raining altogether.
Temperatures are never 'hot' in the Everest Region, but September is a comfortable time to be there, with temperatures dropping to below freezing only at the higher altitudes (that is, the closer you get to Everest) and only at night. Namche Bazaar, for instance (11,290 feet/3440 meters), has an average low of 42°F (6°C) in September, and an average high of 59°F (15°C).
Crowds & Costs
While September is the start of the busy autumn season in Nepal, trails don't get really crowded until October. Most accommodation on the trekking routes re-opens this month after a monsoon break, and it's still not too hard to get a bed in the best places.
An unfortunate fact of life (and travel) in Nepal is that the monsoon causes landslides, which can block access to remote areas. However, the trekking trails in the Everest region are as well developed as they come in Nepal, so by sticking to the more popular routes, you're less likely to encounter slips—and when you do, they're more likely to be cleared away reasonably quickly.
Although lingering rains in September can cause delays in flights from Kathmandu to Lukla, this is still a somewhat more reliable option than taking overland trails (such as the Pioneers' Route from Jiri) that rely on road access to trailheads.
While we are always keen to recommend that travelers venture off the beaten paths and find alternative routes in the Everest region (as well as elsewhere in Nepal), for the above-mentioned reasons, September is one month when it's wiser to stick to the better-trodden trails. The classic Everest Base Camp trek (EBC), shorter treks to Namche Bazaar and around, or the Gokyo Lakes trek are good options for September.
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What to Bring
Many people find that umbrellas are more convenient than full-on rain clothing for showery conditions, especially at lower altitudes where the temperatures are likely to be warm and wet. Of course, a light raincoat is a good addition to any trekker's luggage.
Carry plenty of spare socks so you don't have to trek with wet feet, and ensure that you keep at least one change of clothes for the evenings only, so you're not sitting around in damp clothes. Light wool (such as merino) is good because it stays warm even when wet.
As during all other seasons, it's a good idea to bring your own sleeping bag on the EBC trek. While teahouses usually provide decent blankets, these are not always especially clean, and you'll be especially glad of the extra warmth at higher altitudes. A two- or three-season bag is adequate for September treks.
Getting There & Away
The quickest way of getting from Kathmandu to the Everest Region is to fly to Lukla. While it's important to leave buffer days in your itinerary at all times of year to avoid missing an important international connection, this is particularly important in September. Monsoon rains can linger well into the month, possibly delaying flights to and from Lukla.
Events in September
Nepal's biggest annual festival, Dashain, sometimes falls later in September. Dashain is primarily a Hindu festival, and the inhabitants of the Everest region are mostly Tibetan Buddhists, so Dashain isn't celebrated with as much zeal among the Sherpas as among some other Nepali ethnic groups.
If passing through Kathmandu during Dashain it's important to make preparations. On the most important days of the festival, much of the city closes down, with fewer restaurants and shops open, and taxis being harder to come by. Think of Dashain like Christmas Day in many Western countries—essential services still run, but it's more tricky finding non-essential ones.