Think also about the other months.  While each season is approximately four months, the other seasons seem much longer than summer does.  Why?  Probably because the warm weather invites people to get out and do things to pass the time like boating, hiking, biking, fishing, camping, and the like.  In this manner, the time slips away from us and seems very short.  It is also the time when students and teachers have a break from the routine of school.  Being a teacher, I know how quickly those weeks pass by...I'll be you do, too.  :)

Likewise, winter, fall, and spring seem a bit longer since they are all a little colder and perhaps are not as inviting overall as summer. 

The "lease" is for the amount of time...a short period where summer is concerned...that we have the season.  Like the lease on a car or an apartment, we "borrow" those items and pay for the time we have with them.  The speaker in this sonnet is saying that the time we have with summer is "borrowed" time...it will go too quickly if you don't watch out!

Like a real estate lease (rental agreement), summer has a specific term (time frame). Time is an important theme in this sonnet.

According to the Oxford English Dictionary, lease came into English from the Old French word lais (or leis), a noun form of the verb laisser ("to let, leave"). The Old French word came from the Latin words laxare ("make loose") and laxus ("loose, lax). (The OED is an excellent source for finding out a word's etymology -- it's source, history, and development.)