How much does sea level rise due to sediment deposition? There has been a lot of noise in the news in the last few days because Congressman Mo Brooks from Alabama claimed that erosion played a role in rising sea levels. Quoting from Science magazine:

Brooks then said that erosion plays a significant role in sea-level rise, which is not an idea embraced by mainstream climate researchers. He said the California coastline and the White Cliffs of Dover tumble into the sea every year, and that contributes to sea-level rise. He also said that silt washing into the ocean from the world's major rivers, including the Mississippi, the Amazon and the Nile, is contributing to sea-level rise.
&quot;Every time you have that soil or rock or whatever it is that is deposited into the seas, that forces the sea levels to rise, because now you have less space in those oceans, because the bottom is moving up,&quot; Brooks said.
Duffy responded: &quot;I'm pretty sure that on human time scales, those are minuscule effects.&quot;

Everywhere I have read, people just say &quot;that man is an idiot&quot;; but nowhere have I seen an authoritative, fact-based response to prove this.
How much does the sediment carried into the oceans by rivers contribute to the rise in sea level?
In a 1983 Journal of Geology paper by Milliman and Meade, "World-Wide Delivery of River Sediment to the Oceans" (link) it is estimated that the world's rivers carry about $13.5\times 10^9$ tonnes of sediment per year. If we assume an average density of $2.5~\rm{g/cm^3}$, this corresponds to a volume of 8.8 cubic kilometers. The total surface area of the world's oceans is 360 million square kilometers. This means that the sediment of the rivers contributes to a rise in sea level of 15 µm per year, or 1.5 mm per century.

The IPCC estimated in its fifth report that the average rate of sea level rise is about 3.2 mm per year. That means that about 0.5 % of that rise could be attributed to sediment. It is certainly not a major factor, but is non-zero all the same.