Would you accept a buyout offer that made you rich but you still felt was undervalued?

Personally I wouldn't. I think a lot of entreprenurs get too emotionally attached to their businesses, which is fine if you plan on being around forever and passing it down through the family. For me though, starting a business is about making money and creating new opportunities. A buyout is an stepping stone to my next success.

I was in that situation and turned down the offer. The same buyer came back in a couple days with a fair offer and we accepted it. Had the buyer not come back, we would have been caught in the general market crash and the value of our company would have shrunk. But nobody knew that at the time, and I would not have regretted turning the offer down. I wasn't in a hurry to sell and there would have been other opportunities down the road.

Just like buying a house, you shouldn't get too attached to an offer or buyer until the ink is dry on the signatures. Keep your options open and be ready to walk away even at the last minute; if you don't maintain that perspective you lose control of the process and become vulnerable to last-minute deal changes. Don't let buyers put things in terms of how much money you'll make, keep it focused on what the company is worth and whether it's the right strategic move for the company.

Have you had this conversation with your partners?

This is actually the one sticking point with my partner. He has "mega bucks" on his mind, but if an offer comes through in 6 months for mid 6 figures+, I would be inclined to take it unless we have some serious traction. Not enough to make you rich, but enough to bootstrap you better for the next challenge :)

How much is your pride worth?

All transactions are made on the principle that what you are giving is worth (to you) less than what you are receiving in return. All of them. No exceptions.

If you think that the you are rejecting an offer just because of pride, than it your adding the value of it to the transaction price.

If you do however think that what you have really is worth more than the value offered, you'd refuse.

yes and then off to the races with the next more meaningful endeavor

Unless, of course, you're already working on your life's mission :)