What is the origin of the name "smudge stick"? And why is it considered offensive by some (according to Wikipedia)?
John Trotter Brockett's A Glossary of North Country Words, in Use; with their Etymology and Affinity to other Languages; and Occasional Notices of Local Customs and Popular Superstitions offers an etymology from German:
Smudge, v. to burn without a flame, or any appearance of fire except smoke.—Smudge, or Smush, s. a sulphureous smell occasioned by smoke and dust—close, suffocating air. Germ. schmutz, smut, dirt.
As to offence. While the burning of herbs for their smoke is hardly unique to the American tribes, and is not a matter of cultural appropriation in itself (certainly, it's been common in Europe and Asia since before any contact with the Americas), smudge sticks are portrayed as being somehow a "Native American" tradition. As such, it stands neither as a general example of the worldwide practice of burning herbs for their smoke, nor as an example of respectfully following a genuine tradition of a tribe, but as a mish-mash of the two that often goes against the rules of the traditions they claim to represent. So it could be seen as a bastardisation of genuine indigenous tradition.
It can also be, if not offensive, at least downright irritating to have something that is not directly related to your culture frequently talked about as if it is.
It would also link into wider issues about cultural appropriation. See e.g. Michael Brown's Who Owns Native Culture (Harvard University Press, 2003) or the associated website, for more opinions and information on that.
Of course, many tribes do have a practice of burning herbs for smoke, and I can see little logical reason why it should be argued offensive to use the term smudge as the English word for it (any more than medicine and wheel or vision and quest, though medicine wheels and vision-quests are also recycled into something very different to their original format and purpose, I haven't heard of objections to the words themselves). The difference is likely between those who hear the word most often used of what they really do (or even use the term in English themselves), and those who most often hear the word used of neatly packaged "smudge sticks" that often combine herbs that traditionally should not be used together (it seems some sticks combine sweetgrass and white sage, though one is meant to attract spirits, and the other to be apotropaic, so while there could be a logical reason to use one and then the other, burning both together would be pointless), or that traditionally should not be traded or sold.
There's an irony here. Burning or otherwise using common sage, and other plants called sage (but not white sage, as it is a New World plant) for purification or warding of evil or sickness is a very old European tradition. Smudge sticks which contain common sage (as some do) are therefore a repackaging of an old European tradition with nothing to do with any American tribes. Yet generally smudge isn't used of European burning of incense and herbs, only of American use. So this modern repackaging of a European tradition is being claimed as a modern repackaging of an American tradition, which could seem a bit disrespectful to both.
The Wikipedia article implies that the herbs bound together are burnt to give a scented smoke.
2. a. A heap of combustibles ignited and emitting dense smoke, usually made with the object of repelling mosquitoes, etc. Chiefly N. Amer.
Etymology: related to smudge, v.2
1. b. N. Amer. To make a smoky fire in (a tent, etc.); to fill with smoke from a smudge. Also, to cause (a fire) to smoke; to drive (mosquitoes, etc.) away by smoke. Now rare.
So: no-one knows where the word comes from. As to why it should be offensive, offence is subjective: only those who are offended could explain why.