Re: 1968 Mercury 20hp
About electric starting, possible, yes, but there are some hurdles. The big problem is that these starters are hard to find, no longer available new. If you watch eBay carefully you will see four or five a year, and they will sell for around $60-$120. Two of these sold in the last couple weeks, for about $65 each, within a few days of each other, a coincidence. Look up Item number: 250132190358 and Item number: 250132247269 if you want to see them, a good example of each type.
Also, you need to have the toothed flywheel for the starter. And the other parts (solenoid, wiring harness, etc.). I converted my 1976 to electric start and love it, but I did spend a good bit of time and money to do it.
The starter is 50-47216 but the later replacement 50-97949 will also fit and has some advantages (permanent magnets instead of field windings, so it draws less juice) but I'd say if you find either one grab it. You can also have a non-working starter rebuilt more easily than you can find another one.