Throwing out the Senate would involve re-writing much of the Constitution. I can't find any references to anyone seriously wanting to get rid of it after its inception. Further, bicameral legislatures are very common among governments worldwide.

In any case, here is a good source on the ideas behind creating the Senate and how it and the House changed over the years:

The Framers sought to create a bicameral legislature with different chambers constituted and behaving differently and considering legislation sequentially. While things did not work out quite as the Framers had hoped (the notion of Senators as a new nobility ended in a miasma of corruption that led eventually to a constitutional amendment removing Senate elections from the State legislatures and mandating direct popular election), bicameralism and its benefits as propounded by Madison and others were quickly accepted and hardly ever questioned.