Where did the idea that there *might* be magnetic monopoles come from? I know that we declare that there exist no magnetic monopoles. I see it all the time in my E&M class. $$\vec{\nabla}\cdot\vec{B}=0$$ So there exists no point for $\vec{B}$ to originate from and no point for it to terminate at. We see that the monopole term in the multipole expansion: $$\vec{A}(\vec{r})=\frac{\mu_oI}{4\pi}\sum_{n=0}^\infty\frac{1}{r^{(n+1)}}\oint(\vec{r}')^nP_n(cos(\theta'))d\vec{l}'$$
Is zero. 
But I always hear my professors laughingly say unless you ask a string theorist? What makes some theorists predict the existance of magnetic monopoles? Feel free not do dumb it down if its an advanced topic (im sure it is). It can be fun to lean into ones own ignorance.
 A: I can think of three reasons:


*

*Maxwell's equations (in dimensions where the usual $\epsilon_0$ and $\mu_0$ constants don't show up)
\begin{align}
  \nabla \cdot E &= \rho \\
  \nabla \cdot B &= 0\\
  \nabla \times E &= - \frac{\partial B}{\partial t} \\
  \nabla \times B &= J + \frac{\partial E}{\partial t}
\end{align}
are tantalizingly asymmetric.
It looks like they really ought to be
\begin{align}
  \nabla \cdot E &= \rho_\text{electric} \\
  \nabla \cdot B &= \color{red}{\rho_\text{magnetic}} \\
  \nabla \times E &= \color{red}{J_\text{magnetic}} - \frac{\partial B}{\partial t} \\
  \nabla \times B &= J_\text{electric} + \frac{\partial E}{\partial t} \, .
\end{align}

*Furthermore (as noted in another answer) it can be argued that the existence of magnetic monopole would "explain" the quantization of charge.

*Finally, note that in fact our known theory of electromagnetism does work if we add in magnetic monopole terms (i.e. magnetic charge and current) as long as we also put in the requirement that the ratio of electric and magnetic charge is constant.
That point is argued in the famous electrodynamics book by J. D. Jackson.
A: In a paper published in 1931 Paul Dirac showed that if magnetic monopolies exist then electric charge must be quantized. 
A: The other answers have given good accounts of the initial reasons why we first thought monopoles might exist. However, modern physicists who believe monopoles exist tend to use a different argument.
In 1981 't Hooft and Polakov showed that certain classes of gauge field theory have vacuum solutions that carry a magnetic charge. In plain language, this means that there are solutions to the equations that describe the theory that behave like magnetic monopoles.
The Standard Model, which is the gauge theory that fits best to experimental data so far, is not one of these theories. However, it can be shown that if the Standard Model is part of any higher gauge theory (a Grand Unified Theory), then this theory must admit monopole solutions. Furthermore, we expect that if this is true, the conditions shortly after the big bang would have produced monopoles in copious amounts.
This raises the question as to where all these monopoles are. Some theorists think that this is evidence that no Grand Unified Theory exists. However, it is also widely believed that cosmic inflation may have diluted the abundance of monopoles so there is practically zero chance of observing them in the present day.
There's a good review for non-experts here: https://physicstoday.scitation.org/doi/10.1063/PT.3.3328
And a more technical one here:
http://puhep1.princeton.edu/~kirkmcd/examples/EP/goddard_rpp_41_1357_78.pdf
