The "some reason [it would] be better to connect an external battery to the SD board" is that you need to make sure that the controller chip and SD card are getting enough power that it can write reliably.
To figure out whether and what kind of external power you need you start with the specifications. The first clue is in the general product description to which you linked; it says that it has an "Onboard 5v->3v regulator" that "provides 150mA for power-hungry cards." This is a good start, but you'll see below that we'll need further information from a link on the tutorial page.
Linear power regulators give a fixed voltage output (in this case, 3 V) from a higher voltage input. To work properly they need an input voltage a bit higher than their output voltage; the minimum difference between the two is called the dropout voltage. From the schematic we can see that the board is using an LP298x regulator. A datasheet for a (hopefully typical) part from that series indicates that dropout voltage is 300 mV or 0.3 V. 3.7 V is considerably higher than 3 V + 0.3 V, so you're very likely to be ok in that respect with your external battery. (We also note here that the input voltage can range up to 16 V; we'll use this information later.)
The other thing you need to look at is whether you can supply enough current to the board and card that, under all reasonable circumstances, they can function properly. From the original description you know the board uses a 150 mA regulator which means that the designers expect that you'd never need more than that. (And if you do, the card isn't going to get it anyway.)
For simplicity it's easiest just go go with being able to provide 150 mA or more on the regulator's input, regardless of voltage. (In theory you can supply less power on input if you're giving it a higher voltage, because P = EI, but it's not generally worth the time and effort to go there for a hobbyist project.)
Batteries have a discharge rates known as C rates, and unless you know what you're doing (I don't :-)) your best bet is to stay well within the standard discharge rate as specified by the manufacturer. Adafruit sells a typical 3.7v 1200mAh Lithium Ion Polymer Battery; looking at its datasheet we see that the standard discharge current is 0.5 C5 A, which is 0.5 * 1200 mAh / 1 h = 600 mA, plenty more than you need. So if you're using that battery you'd be safe. (I started out by looking at the 500 mAh battery which has a 0.2 C5 A = 100 mA standard discharge rate; my guess is that that's fine because the maximum discharge rate is 1 C5 A, but this has implications on battery capacity and the like so I'd ask someone first.)
But it would be nice to use your existing power supply for the card rather than use a separate battery, I'm sure. Your best bet for that is the suggestion from Enric Blanco: use an external power supply that can deliver enough current to both the Arduino Pro regulator and the SD Card breakout board regulator that they'll both be happy.
Sadly, Adafruit's Ardiuno Mini product page gives no indication of current requirements. However, a quick look at the v5 schematic we can see that it's using an LP2985AIM5-5.0 regulator and a web search turns up a product page for this that says it's rated at 150 mA.
Add the 150 mA for the Arduino Mini to the 150 mA for the SD card board and you can figure that any external power supply that is giving more than 300 mA at the minimum 7 V input voltage specified for the Arduino Mini should be just fine. The 7 V should also be ok for the SD card board regulator since, even if it's a slightly different model from our spec. sheet, it's unlikely to have a maximum voltage hugely below the 16 V for that part. But you could check the exact part your board is using if you're worried.
