Using some website, the cheapest Tokyo→Paris flight has a 5-hours stopover in Dubai.
I would like to make this stopover 7-days, so that I can visit around that place too.
I can't express this requirement on any of the websites I have tried.
Tokyo→Dubai + Dubai→Paris is almost twice as expensive.
What is a good way to get this week-long stopover without paying much more?
Should I go to an agency? Are there websites that have this feature?
This is one of the cases where a reward ticket (using your points) can carry great value. Most airlines allow one stopover or one open jaw, and some offer more (I've seen two stopovers and an open jaw for some Air Canada redemptions) for the same points. This means that my Toronto to London, ten days, to Venice, three days, back to Toronto trip cost exactly the same points as Toronto to London for ten days and then home again would. (Both London and Venice are in the same zone/area which is how most airlines determine the points to charge you - North America to Europe is all the same.) You generally can't get that when you're paying with money.
I see two different questions.
A simple stopover such as this can be found and booked using the multi-city itinerary option on consumer sites like Hipmunk or Kayak. For more complex searches across multiple dates and connection points, I prefer to use the ITA Matrix Airfare Search, although ITA is not a travel agent and you cannot book tickets through them— you need to try to find the fare available on another website that handles bookings, like the airline's website or something like Orbitz or Cheapflights.
No such flight may exist.
First, there are a fairly large number of airlines that can route you between Tokyo and Paris, especially with a connection. There is considerable competition, and therefore airfares are kept in check. Not every airline flies between Dubai and both Tokyo and Paris, however. With less competition, all else being equal, you will find airfares much higher.
Second, all airline tickets are not created equally, even if your seat and service may seem identical. Your travel is governed by the rules of the fare you purchased. Some fares may allow a stopover at no extra charge. For example, Icelandair will generally allow you a stopover between North America and Europe at KEF (their owners, a tourism company and an sovereign investment fund, have an interest in bringing tourists to Iceland). Other fares may allow a stopover for an additional surcharge. But other fares, especially the heavily discounted fares, may not allow a stopover at all. You would need to buy a more expensive fare or two separate fares for PAR-DXB and DXB-TYO, and that is the reason you get the pricier results.
So, look up the fare rules for the flight you have priced with the short layover. If the fare rules permit a stopover, you can work with a travel agent or with the airline to find availability for your preferred dates. If they do not permit a stopover, you have no choice but to purchase a more expensive ticket or purchase two separate tickets.
Workarounds include using frequent flyer miles, as noted in another answer, but seats may be difficult to come by unless you are flying very soon or have time to check often for flights in the future. Also consider that on most international itineraries, any stop of under 24 hours is considered a layover rather than a stopover. So, while a single day is obviously not as nice as a week, it would still give you enough time to get a glimpse of the city. I arranged a 22-hour connection IAD-JNB-DAR, which provided enough time for a quick tour of Soweto, the CBD, and some other parts of the city.
Ideally, you should find a fare from an airline which allows free stopovers.
The best way to do it is to check the fare rules with ITA Matrix or call the airline or actually test online if the price stays the same when you construct a multi-city trip.
When checking the fare rules we are looking for: FREE STOPOVERS PERMITTED
Here is a handy (but not exhaustive) list of airlines offering free stopovers on most international routes and fares: https://www.flystein.com/free-stopovers/
Disclosure: I am a co-founder of Flystein.
