Which verb to use with "The variable [verb] a value"?

All three terms are used but they are used in different contexts.

For example if you are talking about a variable which is defined as a constant or is an initial condition whose value is determined by the environment before the discussion starts we would say something like "C has the value 10"

If on the other hand we were talking about a set of different scenarios we are likely to say things like "When we double the input voltage then A (the amperage passing through R) takes the value 20"

However when we are talking about the relationship between an input variable and the resultant value we often say things like "If y = 2x then if x if is 4 then y is 8".

There is no definitive answer to your question as the correct (or perhaps the most commonly used) verb changes with the context.

In terms of English, any of your suggestions ("is", "has", "takes") is reasonable. But, as a mathematician, I will point out that "assumes" is quite common in the community. For example, see the definition of "Variate" from Wolfram's MathWorld site (emphasis mine):

It is common practice to denote a variate with a capital letter (most commonly X). The set of all values that X can take is then called the range, denoted R_X (Evans et al. 2000, p. 5). Specific elements in the range of X are called quantiles and denoted x, and the probability that a variate X assumes the value x is denoted P(X=x).

However, I will say that the phrase "takes on" can be used interchangeably with "assumes", as well. Indeed, this may be even more common in a much less advanced setting, as in, say, a high school algebra class.