Should I give equity (or cash) to a partner who dropped before we got business going

...and we never signed any founders or vesting agreements? I now know not having those agreements is the worst thing ever and won't do that again.

We are parting as friends friends so I feel like I should throw him a bone for time spent and contacts that I will use as I move on by myself.

What makes sense to offer?

You should treat him as an early-stage advisor, which means the grant should be 1-2%. If you are very good friends with him and he really helped with getting the company going you may add another point or two. The top limit for some incredibly extraordinary circumstances is 5% because that's already a serious investment stake (employee stock option pools are usually set at 10-20% ).

Also, since you didn't have any formal agreements on shares in place, you have to resolve this situation amicably - a lawsuit between friends is never a good thing, let alone when one is trying to build a company & the other one is trying to take it away.