In order to unlock a Gold Daily Challenge badge, you need to earn between 11000-12000 coins in Daily Challenge mode (the goal will vary by month). Coins can be earned in 4 ways.
Easy challenge (100 coins)
Medium challenge (150 coins)
Hard challenge (200 coins)
Complete all 3 Daily Challenges (150 coins)
In other words, play every challenge for roughly 20 days in each of 8 months.
There are occasionally Expert challenges. These are worth 300 coins, and will speed things up a little, but they are not occurring every day. If you have Minesweeper Premium (the paid version, which is $1.49 a month or $9.99 a year), coins can be earned at double the rate for challenges, but the bonus will not increase.
Sometime in September 2013, an update was made to the game that downloads the Daily Challenges for each month. However, this will only occur if you have entered that month while the challenges were still on the server. You are able to set your PC clock back and retrieve old challenges. At present, it seems February 2014 is the furthest back you can go. If you have entered daily challenge mode during any month and downloaded the challenges, however, you can go back whenever you like. This is why some people can still play challenges as far back as September, while some are stuck with more recent months. This appears to be true for all of the Microsoft Arkadium releases.
This achievement appears to be retroactive. Minesweeper has been tracking medals earned since the release so it is possible to have this already from 2012 and 2013.
(Thanks to Ironspleen
for info on the system clock. Credit to RadicalSniper99
for pointing out the Expert challenges, and Kugare
for the Premium info.)
(EDIT: Updated in March 2014 to reflect testing on the "set your clock back" methods)