I'd stay downtown if you're getting a good deal on the hotel and are only going to Burnaby for the one day. Downtown to Burnaby, is 20-25 minutes(depending on where in Burnaby).
If your hotel is close to BC Place, you should be good. But if you got a young one, the places you said you wanted to visit(Granville Island/Stanley Park) aren't really walking distance, especially in February. In the summer, it'd be a nice walk, but if you're dealing with rain/wind, it could get a little dicey. But a cab can get you to there for pretty cheap. Would be a lot better than driving in from Burnaby or Coquitlam if you plan on spending lots of time in the downtown core.
May I also reccomend heading up to Grouse Mountain, even just for lunch or dinner. The kid will enjoy the gondola, it's a pretty cool trip when you're a kid. The views on a clear day up there are truly amazing.
It's too bad you're coming in February though, there's just so much more to do in the spring/summer. I hope you get some good weather.
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