In RSA encryption, does the value of e need to be random? I am a novice programmer and am just finishing up an RSA encryption program that I am writing for practice. Currently I have the program generate a relatively small random value for the public key e. When adding the finishing touches, I realized that there was no point for e to be random. Is this thinking correct?
Yes, this thinking is correct; there is no requirement that the public exponent $e$ to be random.  After all, it doesn't matter whether $e$ can be guessed by an attacker; we'll be including that value in the public key anyways.

Common practice is currently to use the fixed value $65537 =2^{16} +1$ for $e$.  Any odd value of $e &gt; 1$ will work; however, smaller values of $e$ will tend to make the system brittle against errors in performing the RSA padding.