What should be the nomenclature of this organic compound?
Although this isn't disputed by anybody here, let's start by noting that in general, suffixes have greater &quot;priority&quot; over double and triple bonds in numbering. The rules are given in Nomenclature of Organic Chemistry: IUPAC Recommendations and Preferred Names 2013 (Blue Book):

P-14.4
When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:
[...]
(c) principal characteristic groups and free valences (suffixes);
[...]
(e) saturation/unsaturation

This means that the sulfonic acid suffixes should obtain the locants 1 and 2, which are the lowest possible for two adjacent groups. There are two possible ways of doing this.

With the sulfonic acid groups sorted, we can now turn to the thorny issue of the alkene. It is more complicated than I thought, actually. First, let's consider how to refer to the double bonds in the names. Again, the answer is provided by the Blue Book.

P-31.1.1.1
[...] Only the lower locant for a multiple bond is cited, except when the numerical difference between the two locants is greater than one, in which case the higher locant is enclosed in parentheses.

On the left, this is a &quot;typical&quot; double bond where the locants (2 and 3) differ by 1, and so only the lower locant is cited.
On the right, the locants (1 and 6) differ by  more than 1, so both locants are cited with the higher one in parentheses.

Lastly, we have to make a choice between these two. We turn to the last rule for today:

P-31.1.4.2 If there is a choice of names and numbering, the following criteria are considered in order until a decision is reached:
(1) a minimum number of compound locants. A compound locant is used for a double bond if the locants of the atoms at each end of the bond do not differ by a value of one. When a compound locant is required, the higher locant is cited in parentheses. [...]

Since the option on the left-hand side has no compound locants, and the one on the right has one compound locant, the left choice is preferred. Therefore, the preferred IUPAC name for the molecule is cyclohex-2-ene-1,2-disulfonic acid.