I haven't thru hiked, but I have section hiked some, and there are costs besides food and hostels/hotels to take into account. Little things like laundry, possibly shuttle costs depending on if you can hitch a ride back to the trail or not, first aid & personal hygiene items you may need to buy, postage to mail a bounce box ahead and items you no longer need (i.e. winter clothing in the summer months)home, phone cards, replacement shoes/boots, socks, water treatment, etc. This does not take into account gear that you find doesn't work well for you on the trail & which you may end up replacing, which many thru hikers end up doing. Even shake down hikes don't always reveal problems that can crop up after many weeks of hiking.
Even if you avoid meals in restaurants, also realize that food might be more costly in some of the areas where the only store available might be a convenience store, and after a month or so on the trail you will probably develop the famed thru-hiker appetite, so you'll probably have more food cost then.
Also, the number of nights you opt to stay in lodging of some sort other than a shelter may be determined by the weather and the time of year you start. It's a lot easier to decide to stay on in town when it's cold and wet than if the weather is nicer.
And finally, don't forget lodging in the White Mountains is pretty costly, and you have to stay in the designated areas. From what I've read, only two thru-hikers can do work for stay at the huts and the huts are $80 (maybe more now) per night. As I understand, there are also some shelters or camping areas, but they also charge a per night fee ($8-$9 in the books I've read).
While it would be nice to have money left, it's good you have the option of spending more. It would be a shame to try to do it on such a budget that you didn't enjoy what is a once in a lifetime experience for most folks.
I have a friend who attempted a thru-hike last year (had to leave the trail for family matters), and I believe he spent around $3000 by the Shenandoahs. Yeah, he spent a lot of in-town nights, and replaced several pieces of gear.