Second referendum in general
Or would the "Second Referendum" instead be the same choice between Remain and Leave? Media polls seem to suggest this, but I don't see how this is an option.
Any such referendum is internal British politics. They are to instruct (even though the second referendum, like the first, probably won't be binding) British politicians on how to proceed.
If the majority of people now voted to remain, then British politicians can (this is possible for sure) follow that instruction by changing their negotiation position to maintain as much regulatory alignment. Additionally, they can try to negotiate with the EU to cancel its exit all together (if this is possible depends on the EU and its rules). Alternatively, they may try to negotiate a reentry (though that would be complicated because the EU has strict requirements for becoming a member).
Existing solutions seen in other countries
This is of course only possible if there's a proposed agreement in the first place.
Since it's just instructing British politicians what to (try to) negotiate, they could just pick another country and try to get the same relation with the EU as they have. The Express uses this diagram:
Theoretically, these could all be options in the referendum. Basically, any of these options indicate a balance between alignment and divergence from the EU. That doesn't mean these options could be copied 1:1 and be agreed upon, but they can serve as a basis for negotiation by the UK (without breaking EU rules, because similar deals already exist between the EU and other countries).
On interpreting the diagram
The diagram illustrates existing arrangements between the EU and other countries. The more to the left in the diagram, the more integrated the country is with the EU and its policies. Being more to the right indicates a country less integrated with the EU.
The first option is being an EU member, which is the most integrated with the EU a country can be (this option is quite obvious). Underneath the EU flag it indicates a red line, which means that red line prevents this option from being feasible. Indeed, 'UK leaves EU' means it cannot also be a member of the EU.
The second option is a Norway-style deal. Norway is not a member of the EU, but it does accept ECJ jurisdiction, free movement of workers, substantial financial contributions and it does not have regulatory autonomy. If the UK does not accept one or more of these restrictions then the Norway-style deal is off the table.
The third option is the Swiss-style deal. This deal still requires free movement, a substantial financial contribution and does not allow for regulatory autonomy. The difference with the Norway-style deal is that there is no ECJ jurisdiction. Instead, there is another mechanism in place (this is quite complicated, this article by the Guardian touches on that).
Going further to the right, the options can be read in the same way. At the very right there is a line saying no deal. This means that if the UK does not accept any type of concession then a deal is not possible at all.
Speculation on a second referendum
As Hedgehog points out, the contents of a (possible) second referendum are not clear (yet). In that answer, it's pointed out that a high-ranking Labour politician argues the result of the first referendum should be assumed. On the other hand, others within the Labour party have suggested not ruling out remain as an option.
The PM wrote a column earlier this month stating clearly: "There will be no second referendum on Brexit – it would be a gross betrayal of our democracy".