No, you would have to hold the option for an entire year before qualifying for long term capital gains.

It works the same way as with stocks. If you liquidate your position then repurchase the same security, the holding period resets.

According to page 56 of the 2015 IRS Publication 550 on Investment Income and Expenses:

Wash sales. Your holding period for substantially identical stock or securities you acquire in a wash sale includes the period you held the old stock or securities.

It looks like the rule applies to stocks and other securities, including options. It seems like the key is "substantially identical". For your brokerage / trading platform to handle these periods correctly for reporting to IRS, it seems best to trade the same security instead of trying to use something substantially identical.