For me it depends on a lot of factors, as I'm sure most folks will say.
First thing I do when considering an offer is go to Trulia to see the price history on the house. Additionally I'll go to my county website and look at sales of the house and its assessed value.
Next I'll chat a bit with my contractor about general costs to rehab (although I'll have a good idea of the rehab costs after looking at the house).
So with those two pieces in place, I can get a good idea of the ARV and what I would offer on the house. But there are some additional things I consider:
Generally speaking, the longer the house has been on the market the lower I'm willing to go with my offer price. Two schools of thought on this are:
1.) Nobody has offered on this house and they would be more willing to take ANY offer.
2.) The second school of thought is they've already rejected all the low ball offers so don't waste your time.
Finally when considering an offer I'll trust some amount to instinct. In the situation I referenced I had a good idea that not very many (if any) folks would be offering on this house. This house is just two away from one I already own, so I asked my renter to keep an eye out for how many folks were in/out of the house and he didn't see anyone go in/out during the bid period.
It wasn't a make or break deal for me. If I got the house, great. If not then move on.
So on this house offer price that was accepted was 40% below asking price. I offered all cash with no contingencies and close in two weeks.
Again, for me it all depends on how well the house fits into my long term goals whether or not I do a low ball offer or not.
So far I've only purchased 3 homes. I've not been at this very long so I've a lot to learn. But of the three homes I've purchased I've paid 10%, 25% and now 40% under asking price. I kind of like where the trend is going :D