I agree. Go for the strings first. As strings age they develop more overtones. One way you can see this is with electronic tuners. As they get older the note on the tuner appears to "bounce around" the target note and is harder to dial in.
The overtone when fingering the B note on the G string could very easily be sympathy overtones produced on the actual B string.
Another, less common source can be either the saddle or nut. If either does not have a good edge then some unwanted contact can occur. Check the saddle to see if it is rounded too softly at the top. If the curve is not sharp enough then the vibrating string can be making contact just in front of the saddle edge and producing something that sounds like an overtone with a slight buzz. Same thing can happen on the nut end. I see both of these conditions mostly when inexperienced people try to make changes to them.
My saddles use an edge with very little rounding. In fact it can be hard to tell its there unless you look close.
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