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Imagine you were renovating your bathroom and it turned out after they ripped down the drywall that there was a bunch of rotted wood and stuff. The work would end up costing you a little more, and you'd accept that because you want the job done right and you don't want rotted wood behind your new bathroom walls. The same goes for programming. If he underestimated the time - sure, it's his fault - but it's the quality of the work that will suffer. Everything will be done half assed if he works out that he's doing it for $6/hour at the end of the day. Taking a "nope, you said this much and that's all i'm paying" won't help you out much when there's a bug weeks down the line that needs fixing and he starts ignoring your emails. I'm not saying pay him double, but if it's a $1500 project then $1800-2000 won't kill you and it will make a difference in your end project and it will make him happier - this is a person you will need to keep contact with for the foreseeable future.
Keep in mind the amount of work it is might have changed... You might be creeping in features, changes, etc. Move this over 10 px, change this padding, can you make this work like this instead, etc all takes time to do. A 10 minute change still might take 20-30 minutes when you factor in communicating back and forth messaging you "ok check it now" and so forth.
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