Bummer, but all is not lost. This is actually a good thing to happen now, rather than later. How much investment? A lot? Can you do without it? Can you repay them back? Is it worth keeping someone on who doesn't really want to contribute but still wants some control?
'... we have been working on the project for a few months ...'
This is good. The longer you work on the project the more investment is required (time, money and effort). It's also good to weed out early someone who is not willing to go the full journey. So in this respect you are lucky. You haven't created a lot of value that has to be split up or given a dollar value.
'... What to do when your co-founder takes a backseat? ...'
Use the ejector button?
'... What do you reckon I should do? get another co-founder? ...'
Find another co-founder. Work out if you need your ex-founders investment, skills at a later date. If not re-pay the investment. Try to determine why the co-founder has left? Is it because of lack of interest, skill, commitment? Is it they found a better offer?
1) The investment is not a lot in monetary value but I couldn't get started without it, in that respect it is "worth it" I could probably pay the person back if the company made even a little money. The person is still involved in the sense that I could ask for help and ideas etc but i would not expect proactive participation.
2) You are right, almost no value has been created as of yet, so if we were to shut up shop tomorrow, he would have lost out.
3) I think the ejector button would be too strong, the advice etc this person has given and i am sure will give will be invaluable. however, i would not expect a LOT of work in terms of execution. But then execution is what matters. If the company did well I believe this person would join. But i would not expect him to leave his job in uncertainty.
4) He left because he is just not THAT interested in Web Apps. No interest = no determination and no determination = no startup in all honesty. Maybe I should have seen this coming. I was visibly more enthusiastic than him.
I think I will push forward until launch (mid-summer) on my own but will look out for another co-founder. If anything this situation has made me more determined to succeed. I am reminded of Evan Williams at Blogger!
1. What he/she gets as part of investing capital in the company (and whether that's going to be equity or debt)
2. What he/she gets as part of having an operating role in the company.
The good news is that this happened now (vs. later). The bad news is it sucks to lose a co-founder.
If this is the case I'd repay money and take the person on as an early employee but not a co-founder. This is okay. You know the person. But the fact is they did not think the idea, concept or work compelling enough to quit. Also don't be fooled into thinking early investment is some sort of ownership.
'... I think the ejector button would be too strong ...'
That was meant to be a joke
If you need their investment, negotiate based on it as an early investment rather than as a partnership (a partner gets lots of sweat equity and a pittance based on some small amount of cash invested--equity bought with cash is usually much smaller and on less favorable terms in the sense that small sums do not bring seats on the board, etc.). But you probably don't need that money. Buy back their share (if you've already incorporated and issued shares), and call it a friendship rather than a business partnership. You do not want to deal with these questions down the road when the company has assets.
I don't mean to be patronizing (I don't see how it remotely is). Going at something with a partner isn't a requirement for success, nor is it only for "cheering up." Investors like to invest to successful ventures, and a business partner is a nice assurance for them.
But go ahead and show 'em, you rebel without a cause. ;)