I believe you should use your free time the way God leads you to use it.
For me, I have done both resting and sightseeing on my down times during mission trips.
Best advice? Pray about it. There is a chance that Holy Spirit was trying to convict you using your friend. There is also a chance satan was using your friend to try to steal your joy. I have been in both situations. Pray and ask for guidance. If you are supposed to do some sightseeing, the doors will open for that. If not, chances are they wont. Ask to be sensitive to Holy Spirits leading and you will be fine.

I believe there is wisdom in both of your points.  If you are truly going over to do missions, then your work is tremendous.  I'm assuming you are supported by a Church and you don't want that money to go for just plain old sight-seeing.  That wouldn't be responsible.

However. You're human.  You're Ministering to a certain kind of people in a certain culture.  You need to know that culture and how they think, their history, and what has shaped their country, mind, and heart.  Exploring their country will also give you chances to reach out to people beyond your local area.  Perhaps if you can find another missionary couple you can coordinate your exploration with a chance to speak or teach or encourage where they are working.

I, too, am about to go on a mission long-term.  To me, it is about mission work.  But I DEFINITELY plan on doing "sight-seeing".  For the reasons I listed above. 

So my point is that, yes, it is all work and no play; but that doesn't mean you can't sight-see and engage in other leisurely activities.  You just need to have your "head up" and ready for every opportunity that comes you way to use it for Gods glory. 

I am currently on a two-month mission trip in New England and one of my best spiritual conversations so far came from spending a day at the beach!  God will use you wherever you are if you are willing.

Like others said, it depends on your real purpose.  I'm a missionary in Northern Ireland (I'll be here for at least two years), and while I haven't had much time off yet, you'd better bet I'm doing some exploring on my days off!  Has your friend done mission work before?  If all you ever do is work, work, work, I can almost guarantee that you'll get burned out, and you won't be as effective.  You have to have personal time for study and prayer.  In my experience, you feel closer to God while out exploring His creation.  That closeness is integral to mission work.  Also, you have to take care of yourself.  You can't help others if you haven't helped yourself.  That doesn't mean to indulge in things.  Make sure you take care of your own needs.  It isn't selfishness.  And sometimes, alone time / relaxation time is a NEED.  

i've been on several mission trips. as far as free time goes, i think a balance between work and play is good, just as it is with anything you do. plus, the play can be used to bring the group closer together, thus it could be considered work to build the Kingdom, and the work, if the group can be encouraged and motivated to serve and enjoy doing it, could be considered play as well. in fact, on one of the work days during a trip, it ended up raining really hard, so we couldn't really do much work. for some reason, instead of heading back to the sleeping quarters, which we affectionately called the hollow, we stayed at the site under an open air garage and played games like psychiatrist.

additionally, and this is just a tangent, but who says missionaries have to go to some far away place to do their mission work. i currently work as a missionary serving the campus of my alma mater through the local Catholic Church. it's only my second week there, but i'm doing a lot of prep work for the school year. i love it. just thought i'd point out that even the people we see everyday need evangelization!