Does neuron have weight? My understanding is that a connection between two neurons has a weight, but a neuron itself does not have a weight.

If connection c connects neurons A to B, then c has a weight w, but A and B don't have a weight. w determines if A has a strong influence on B or a weak influence on B.

But this Wikipedia article says:
"The connections between artificial neurons are called ‘edges’. 
Artificial neurons and edges typically have a weight that adjusts as learning proceeds."

https://en.wikipedia.org/wiki/Artificial_neural_network

The 2nd sentence seems to imply that a neuron has a weight, too.

Does a neuron really have a weight?
If you look at the 2nd equation under "Propagation function"

$p_{j}(t)=\sum _{i=1}^No_{i}(t)w_{ij}+w_{0j}$

The $w_{0j}$ is the constant (bias), so it's reasonable to write that the $w_{ij}$ weights are associated with the connections, and the $w_{0j}$ is a weight associated with the nueron itself.

Additionally if you assume $o_{0}(t)=1$ then you can rewrite the equation above as

$p_{j}(t)=\sum _{i=0}^No_{i}(t)w_{ij}$

(i.e. note the sum is from 0 to N, not 1 to N)

When you do this it no longer really makes sense to make a distinction between a "weight" and a "bias".