One habit that I have found useful, and for me came from making (maths based) art rather than mathematical training is to think in terms of aesthetics rather than mathematical correctness when exploring a problem. Take approaches that feel interesting, exciting, beautiful even if you also know that they are wrong. On occasion this can lead to the broader understanding needed to either generalise the problem correctly, or simply suggest an unusual (but correct) approach that might not have been considered directly. For me this often takes the form of trying to make images related to a problem, with the only question being whether an image is visually interesting or not.
I think that it is quite possible to come to this technique from within mathematics, but for me it came from outside, so it seems relevant here.
Thinking about this a little bit more, an additional habit is to seek a historical perspective. Looking at how problems and questions developed and gaining a sense of what ideas have been used to approach them in the past. I would argue that this is justified simply to grow an appreciation of the culture of the subject; however it can also bring benefit. Getting a sense of different approaches broadens the number of ways that you can attack new related problems where an outdated technique might suddenly find itself useful again!