Budget is around $200 & there are 3 of us. Decent options for a morning coffee also good!
Any hotel near Te Papa is in the right area for you - that's where the food and coffee options are to be found. You could look at the West Plaza or Bay Plaza or, if willing to go above budget for something a bit more special, the QT Museum. A little bit further away from Te Papa but more central there is the Mercure Wellington Central City.
Almost all of the city centre hotels would be convenient for Te Papa and there is good coffee within a short walk from anywhere in Wellington. The Amora or the Museum Hotel are both close.
If your hotel is close to Courtney Place, I recommend the Airport Flyer to the airport, esp if it is peak hour. The buses use a bus only tunnel under Mt Victoria as well as dedicated bus lanes, so get there very quickly.
We have stayed at Bay Plaza several times. It is a bit 'retro' and not upmarket and shiny like a new hotel but the view is outstanding. Look online and you may get the $110 deal per room which is very reasonable. There are lots of restaurants and bars very close by.
They offer a $25 taxi fare to the airport (or they did in the summer it may have gone up by now).
Great - thanks for all the replies. I'll check out the hotel recommendations for sure.
FYI it's likely we will have a hire car to return to the airport when we leave, so I assume we will want hotel parking.