The only way for you to convince these people, is for you to show them that there are a new untapped source of income for them. I did this with a pub and I generated repeat business by hosting some
Bitcoin meetups at this pub. We gather there every second week and we only pay with Bitcoin. The worst thing that will happen, is if you convinced him/her that Bitcoin is this wonderful new currency,
and after 1 or 2 months he only received like 10 to 100 transactions. So once the door is open, you have to make sure people know about it, and they use that service or merchant regularly.

Surprised to see that no-one has replied to this (unless I missed it), but this really is very important if you're going to convince a business to accept Bitcoin.
If you suggest accepting Bitcoin to a business and they try it out and they don't generate enough revenue with it, than it's actually quite damaging to the image of Bitcoin and they would probably not try it again if it fails the first time.
I would suggest that if you do manage to convince a store, pub or restaurant to accept BTC, you at least go there yourself and pay with Bitcoin regularly and tell all your friends about it.
Also, go the extra mile and add that business to any website or service that lists businesses accepting Bitcoin.