The HSA contribution limits for a partial year can be very confusing. This is discussed in IRS Publication 969. There are two different ways to determine what your limits should be: the prorated limit and the last-month rule.
The standard way to handle the contribution limit for a partial year is to prorate the limit for the number of months that you had HDHP coverage. There is a worksheet on the instructions for form 8889 that walks you through this, but essentially here is how it works. For each month, you determine what kind of HDHP coverage you had on the first day of the month. In 2015, if you had single coverage, you would enter $3350 for that month, if you had family coverage, you would enter $6650, and if you had no HDHP coverage, you would enter 0. Add up all 12 months and divide by 12. This is your contribution limit for that year.
In your example, where you had single HDHP coverage only on December 1, 2015, your contribution limit using this method would be $279.
Last Month Rule
If you want to contribute more, you may want to use the last month rule. The last month rule states that if you have HDHP coverage on the first day of the last month in the tax year (December), you can contribute the entire annual limit for the year ($3350 for 2015). However, there is a catch. If you use the last month rule, you are subject to a testing period. This testing period begins on the first day of the last month (December 1, 2015) and continues until the end of the following year (December 31, 2016). In order to use the last month rule, you need to remain covered by an HDHP for the entire testing period. If you use the last month rule, but fail to remain eligible for the entire testing period, you will need to add the extra amount that you contributed back into your income for the following year. In addition, there is a 10% penalty.
In your example, since you are covered on December 1, 2015, you are able to use the last month rule and contribute up to $3350. If you remain covered by an HDHP for all of 2016, everything is fine. However, if you stop being covered by an HDHP at some point during 2016, you will have to add to your 2016 income whatever amount you contributed to your HSA in 2015 that was over $279. In addition, you'll pay a 10% penalty on this extra amount.