interface arduino with off the shelf generic consumer electronic devices

If I were to buy all the parts needed to build a digital clock, one by one, it's very expensive. But a made in China clock is much cheaper. And and old clock sitting on the side of the curb is free. Now the same holds true for digital bathroom scales, and tons of other things as well.

What is the cheapest way to go about finding where to "tap in" to the original device's motherboard and read this data in the Arduino?

There is no cheap way. Either you have new parts with known data, or you have old parts with unknown working status and you find data for the parts and hopefully find datasheets for them and buzz out the connections on the board, and work out what you can about how itr works.
Time is money - which do you have more of? Do you have a one-off project in mind, or something repeatable? Its all about tradeoffs.

Hi Capt'n,

"What ever happened to the kid who took alarm clocks apart??"

I used to ask that at "Career Fairs".. Glad to see some of you survived.

Do a little googling at "electronics repurposing" or "Computer Demolition"..

OMG, #6 on Google is my 1985 stuff! http://goo.gl/qTSSg My Favorite part (Probably 'cause I wrote them ..):

”Will American kids be Inventors in 10 years, or just operate appliances designed elsewhere?”
and, by the way,
”Whatever happened to the Kid who took Alarm Clocks apart to discover how they worked?”

...just get started and Don't Worry About it...

What is the cheapest way to go about finding where to "tap in" to the original device's motherboard and read this data in the Arduino?

Well, in short, practice.

Devices are rarely documented very well within the walls of the design team who made them, so you can imagine how much of that makes it to the public.

So practice taking things apart and analyzing the components / circuit.