But it requires thinking before last year.
http://www.ushistory.org/gop/origins.htm
"The name was partly chosen to pay homage to Thomas Jefferson's Republican Party," wiki adds to the foregoing.
We live in a "Republic" and those who organized the (R) party viewed them as preserving it. Ultimately it's a Latin phrase, "res publica", the "public thing." Some languages kept the 's', some didn't. French lost final consonants, for the most part, and we get our word from Latin through French. Most people have no idea what a republic is and this frustrates them greatly at times. https://en.wikipedia.org/wiki/Republic#United_States will do.
The Democratic Party was originally the Democratic-Republican Party and was the only party for a while when the Federalists went bye-bye. National Republicans came along who wanted more use by the central government of national resources, build out infrastructure to unify the country into a single entity. Think of them as centralizing or unifying. Jackson and the Democrats favored a bit more decentralized view, more agrarian. Much of the history of the US, from 1776 to the present, has been about consolidating power in the hands of fewer and fewer people. The Democrats were later the "states' rights" folk. Both parties are very pro states' rights and also very anti states' rights, but that flip-flops by issue.
National Republicans went away, becoming Whigs. But the Republican-Democratic Party had lost the "Republican" part of its name in common usage, and that became official in the 1840s. You seriously can't contrast "Republican-Democrats" with "National Republicans." Too much overlap, not enough of a clear division between the Good and the Evil, esp. when both sides see themselves as the Good.
Republican Party came along in the 1850s, apparently in a response to all those Democrats coming up from the South who supported slavery but it quickly spread. And it was, again, centralizing. Remember: Lincoln was a Republican. Southern "blue dog" Democrats were the traditionalists in the party. The Democrats' centralizing tendency came along strong later when things shifted from asking "where does power for a person to act independently reside?" to "what should be done for whom?"
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