What is a thorough method to manually generate a random number between $1$ and $100$?

Take your paper and cut it into $100$ smaller (equal sized) pieces. Write each of the numbers $1$ through $100$ on the pieces. Then mix up the pieces and pick one at random.

If you don't want to do $100$ slips, use $10$ slips, labelled $0,1,...,9$. Pick twice with replacement for the two digits. If you get $00$, that's $100$.

A method I've always liked is to put down a bunch of parallel lines on paper and pick one somewhere in the middle, and count how many lines are on the left of it, and take that $\mod n$. The number of lines you start with of course depends on $n$ (it should be a lot more than $n$) but this is okay, since we don't need very high numbers $n$ anyway. In the case of $100$, we take $n=2$, $n=2$, $n=5$ and $n=5$, and construct the random number from these results.

You can find printable paper dice templates on the 'net:
http://www.timvandevall.com/templates/printable-paper-dice-template

If you don't a model with as many sides as you need,
you can always create your own in Blender,
then export a paper model by way of this pepakura script:
https://wiki.blender.org/index.php/Extensions:2.6/Py/Scripts/Import-Export/Paper_Model