Purely for web development? Or does future employability have any bearing on the decision?
Almost purely for web development - possibly Node to get thinking at the HTTP level and understanding what is actually happening beneath the page.
Can we assume that you already know html? It's the foundation. Think of it like knowing how to add and subtract even if you work in the far reaches of higher mathematics where you never meet anything so prosaic as a number. Some things you simply have to know.
And get comfortable with Regular Expressions. A healthy respect is OK-- you can seriously injure yourself with a carelessly constructed RegEx-- but you will meet them everywhere you turn. So try to internalize the basics.
I'd suggest getting a handle on at least one server side language, such as PHP. That goes very well with learning a bit of MySQL and once you've got the two skills combined you're straight away employable (to a certain degree!).
Granted, you're probably not thinking about employment at 13, but it's good to have a long term view!
As lucy24 has said, this assumes you've already got a handle on HTML/CSS.
If it was my 13 year old,I'd find them something to do outside. They're probably already spending too much time in front of the screen.
If they can't find anything to do, the lawn needs cutting again.
I started off learing basic when I was around 13 or so... then took a long break from computers and got back into it around 93 or so.
The frustrating part has been how you learn one language and then it seems to become "obsolete" by the time you master it.
I suppose a better question would be - what to focus on that won't become obsolete for the next 10 years or so?
Heck, these days MS offers so many avenues (and should be taken by 13 year olds) that it's a no-brainer. Google is, too, though that "open source" is about as closed as the door on a convent. :)
Any "language" studied will provide the intellectual tools to study other languages. But at 13, one needs both nudge and wisdom as to future possibilities and the market place. It is the last element that is dang difficult to understand at 13!
I would go with Python + Postgres
Python because there is less stuff all bundled into the name space when compared to PHP which has a bloated name space.
Postgres because it works more like Oracle and other higher production databases.
MySql misses things that work everywhere and implements tricks that aren't used anywhere. Auto_Increment is an example of the latter.... not having the ability to set functions as a default value is an example of the former.
I will echo Tangor's sentiment that it is best to learn to program. The language doesn't matter, it is just a matter of sytax at that point, so I would say teach them to program, then give them a language.
I still like Python for starting out as the name space is light. For example, I feel like it is a better path to learn to code in a method that sanitizes string inputs for DB inserts/updates than to teach them to wrap it in mysql_real_escape because that doesn't teach them how mysql_real_escape works.
I'm also starting to think one may be better off at mastering one area like jQuery and focus mainly on that. The competition is very tough out there and I wonder if hyper specialization is a better strategy?
The trouble is that in general as a web developer you may be expected to cover off a number of languages/skills.
Not saying there's anything wrong in specialising, but that's not necessarily what employers are looking for.
And even if you are considering self-employment, being a jack of all trades may be better than a guru of one.
Just my 2c