Invest whatever you feel comfortable with losing. Never mortgage the house or put your kids at risk.
Do not worry about what impression your cash will make to future investors. The dirty secret of VCs and angels is that it makes no impression. What matters is how much traction you are able to build. No one ever really cares that you put $100K from your own savings, or $1M, or $5,000, as long as you have the next twitter.
Determine how much you can spare, and invest that.
If you start to run out of funding, or you know you need more to finish then you will need to try to bring in investors.
But, the farther along you are the easier, I expect, to get investors, if your only issue is money.
If you can spare all in the account, then it is up to you, if you want to roll the dice, invest it all.
I have seen people mortgage homes to help with funding, but to me that is too high risk.
I prefer the slower, safer approach. Get most of the development done, and then if I need help with bringing it to market then look for an investor.
Most investors will want you to have invested significant time and money into your business before they even think about investing themselves. It's a demonstration of your commitment.
How much you choose to put toward the business will depend on a lot of factors:
So the short answer is, "it depends".
I think the answer to never mortgage the house is overstated. It can make sense. It is risky and not the best choice, but it could be a good option in certain circumstances.
If you want to avoid angels or VCs and actually take the company to a successful level you might have to use financing.
It all really depends on your comfort level and the prospects of success.
The bottom line is that people generally DON'T want to give you money when you need it, but will throw money at you when you DON'T need it.
Much of it also depends on the type of businesses - some will need outside investment (lots of it) some won't - you have to figure out what makes sense for you and your business.
A one-size fits all answer is not possible and in fact, multiple different solutions can also work.
You need solid answer so I'd call it $10k - $50k
If you don't have a proof of concept for around $10k out of pocket, you should probably have angels on board who can fund what is apparently significant development.
If by $50k you don't have something that can get you more attention and alternative sources of funding, at the very least, a few angels who have said, "Get to X and we're in," I'd say its time to go looking for something else.
I don't have TONs of experience with this but I've been in a couple startups and have started a few basic online companies (for less than $10k)