Warning! Cosmere spoilers (not really though)...
It comes from the assumption that many have made which is as follows:
Fact: Rayse (probably Odium) splintered Aona's and Skai's shards, presumably so no one could take their powers. And he killed Aona and Skai.
Fact: Splinters (like Divine Breath) are non-human objects that hold part of a shard's powers.
So: Aona and Skai splintered into Splinters (assumably Seons and Skaze).
So if we know this, we can go like this:
Fact: Odium killed Tanavast (Honor).
Assumption: Odium doesn't want anyone to take Honor, so he can remain unchallenged.
So: Odium Splintered Honor.
Further So: Honor splintered into Spinters, which may be: Honorspren, Shardblades, Shardplates, Spren, the Storm, etc.
There is also the quote that they wanted Urithuru to be built in the West, to be closer to Honor. This could mean closer to Tanavast's body, or it could mean Honor's pool, or it could be the body of Honor (like Atium?).
I now realize that was mostly off-topic. Oh well.
Endra, it doesn't say in the book (that I know of). But Dawnshards sound like such awesome things, that maybe Honor gave a part of himself to make the Dawnshards (like how Endowment probably gives the Divine Breath).