I play and do sound work with a lot of folks who ask my advice on on-board mics before and/or after buying them, so I've vicariously tried a lot of approaches.
The first reason to pick an on-board mic is waaaay better sound! Compared to any mic you'll encounter at most venues, an on-board mic will sound fantastic. And putting the mic right on your instrument cuts an enormous amount of background noise out of the picture, which is half the battle of live sound.
Another great reason is the freedom of movement you get with the mic on your instrument. Often after starting to use an on-board mic, fiddlers in particular shed some of the muscle pains that can come from scrunching up in a fixed position trying to stay in front of a regular mic. Besides, it's a lot more fun to boogie!
A more subtle reason is less variability in your sound situation from night to night leads to better results and confidence that everything will work the way you expect.
What Mic to Buy
The best thing to go for these days is the Audio-Technica Pro7a, with an Audio Technica 8418 Unimount (mini-gooseneck). It's a good choice for a bunch of reasons: (1) it sounds good (2) it's cheap [$85 for the mic, $30 for the gooseneck], (3) it's very reliable, and so many people use 'em that even if you break it you can often borrow one and be comfortable right away. And A-T still fixes them.
Some people have opted to spring for the Pro15a, which is 2x as expensive. It has 2 liabilities, besides the cost: (1) it has better frequency response - yes, you heard right - which only manages to amplify fingerboard noise into loud thumps and also has too much top end, (2) it can run either from phantom power (supplied from the mixer) or batteries, but if you're running off batteries and somebody hits the phantom switch, the gain jumps by 5-10 times, causing screaming feedback and gnashing of teeth.
Physically the 7a and 15a are identical: very small and light mic capsule, with a long thin lead to a small beltpack (about the size of a pack of cigarettes) which contains a battery and some electronics. You plug a standard xlr cable into the beltpack, which besides going on your belt, can clip to a music stand.
There are two 'on' positions on the beltpack: the one with a straight line is flat, no eq. The one with the bent line is a bass rolloff (it's standard terminology on mics, now that you know it!). If you have the Pro15a, you can use this to reduce a little of the thump.
The Pro7a technically hasn't been available for years, because it was superseded by things with numbers like 831a, etc, which as far as anyone can figure out are identical but with a higher price tag! So if you call, stores may tell you at first you don't want the Pro7a or they don't have it, but they do and you do.
There are any number of fine miniature condenser mics like the Pro7a; however, all those I know of are more expensive, and in a live situation you won't be able to hear any difference.
Where to Buy
Positioning the Mic
For a very few fiddles this position seems to generate a very 'sizzly' or 'grainy' top end. The other spot that seems to work well for these folks is to point the mic almost into the f-hole on the bass side, and often very close to the wood. Try the other spot first - the f-hole location will be very muddy and woofy if it's not right for your instrument.
Mounting the Mic
I think Walter Lenk developed this approach: take a piece of 12-gauge insulated wire (from a piece of Romex, household wiring-type cable) about 10" long, and coil it around the mic element, which is a little cylinder. Then you need to rig that to your fiddle so the mic is pointing at the bridge from the tailpiece side, at about a 45 degree angle, and about 2" away. There's a bunch of ways to do this; some people drill the tailpiece and put a banana jack into it, then fasten a banana plug onto the wire, which then can just plug into the jack at the gig in 2 seconds.
I've done something similar with no drilling, using a pair of #10 washers on the banana jack, and clamping it between the D and A strings. You want the washers, etc. to be as close to the tailpiece without touching as possible. Also, the best banana jack to use is the one from Radio Shack that comes 2 to a bag for $1; it's the smallest, lightest one I've found, and you get your choice of elegant black or rich red. Before you mount it, cut a little off the solder tab, leaving just enough so you can bend it onto the bottom of the plastic part so the internal jack won't rotate after you put the mic wire in. This will be clearer once you have the parts in front of you!
You can just fasten the wire into a banana plug as mentioned above. Or there's the ultra-light fanatic way: take a banana plug, and with a pair of long-nose pliers, screw the plug end around until just the flexible part falls out; hard to explain in text... then solder just this flexible thingy to the end of the 12-gauge wire.
You can use this same mounting strategy with most of the small condenser mics on the market except for the Crown GLM, which is a different shape and much smaller.
P.S. Note that the 12-gauge wire has no electrical function - it's just a support.