Bass Guitar

Bass Guitars are cheap and very easy to find in locale music shops.
You can pretty much walk into a music shop and buy a bass you like.

However, there are a few things I would recommend you look for.

Size

Have the shop assistant try your child on a full size and a 3/4 size bass (sometimes called short scale)
It is important that the child puts the bass on their right leg and can comfortably reach the furthest frets, without awkward posture. Due to the design of the bass, some similar sized basses are easier to reach than others.
For this reason, I am a big fan of the Ibanez SR series, available in 3/4 and full size.

Active vs Passive

Active simply means there is a battery (usually 9V) in the bass, which is used to send a stronger signal from the Bass to the Bass Amp. While active basses are a wonderful idea, I don't recommend them for Primary School students for one simple and obvious reason..... they will forget to change the battery! The most likely time the battery will go flat is right in the middle of the most important concert of the year. There is enough for the young bass player to manage already, including their lead, tuning the bass before playing (every single time they play), setting up the Bass Amp and ensure good level and tone and of course remembering their music.

Simply put, buy a Passive Bass for your young student.

Left Handed

This is a controversial subject. 
If you walk into your local music store, you will likely have sales reps pushing you to buy a Left Handed bass for your left handed student. Remember, very few of these reps are professional bass players and they almost certainly haven't spent hours teaching bass to young students.

The facts as I see them, and have verified with many top bass tutors:
1) when playing bass, both hands require dexterity and control. If anything, the Left hand has the hardest job, so being a Left Handed bass player, it is no disadvantage to play a standard (right handed) bass.
2) buying left handed basses - you don't get nearly the same range and variety of options. Often the Left Handed versions will cost a premium.
3) back up - if for any reason your students Left Handed bass is forgotten, broken or not working, you won't be able to just borrow someone else's.
4) learning by copying - many bass players are strong kinaesthetic learners, which means they learn most easily by doing and copying. Being on a mirror image bass will make this more difficult.
5) moving onto double bass - there is no such thing as a Left Handed double bass (unless it was custom made). Ultimately as a bass player moves into high school, it is ideal for them to progress onto double bass, therefore it makes sense for them to learn correctly from the beginning. Go watch videos of the best orchestras in the world, no left handed basses.
6) there are many good Left Handed bass players who play normal right handed basses. There are some top players who do play left handed electric basses, but this is hardly a good reason for a young beginner to emulate them. These virtuoso exceptions were almost always self taught in the beginning.

Ultimately it's up to you and your child, but my strong recommendation is to begin on a standard, right handed bass.

Ibanez SRM20GB 3/4 Short Scale

(Recommended - for smaller beginners)

Current Street Price: $ 349

IBANEZ GSR200

(RECOMMENDED)

Current Street Price: $ 349

Bass Amps

Your child will need a Bass amp to practice at home.
The quality of Bass Amp often makes more difference to the tone and sound than the Bass Guitar does.
It's not worth skimping on quality, as the amp will be a large factor in the pleasure your child experiences when practicing, and lots of wonderful practice is essential to all band instruments, including bass guitar.

Producing Bass Notes, requires a lot of power and moving a fair bit of air.
For this reason, I don't recommend using the tiny little practice amps which can be found on the market.
You can always turn a larger amp down, but the small ones just don't produce good tone in my opinion.

Fender Rumble 40

Current Street Price: $ 389
This is the smallest Bass Amp I'd personally recommend for home use.

Gallien Krueger MB112

(Highly recommended)

Current Street Price: $ 685
This Bass Amp is amazing.
With a 12" Speaker and 200W of power, it sound truly gorgeous.
This is what I buy for my smaller school bands. For the bigger bands, or when there are two bass players, I add the 15" version of this as well.
The amp is so light it's unbelievable until you lift it for yourself.

Yes, this is overkill for practicing in your bedroom.
But you can turn it down nice and quite and it still sounds fabulous.
If and when you need an Amp to take to a Concert, you are already covered.

By the way, I've also run a Double Bass with a pick up through one of these and it sounded incredible.
GK are the choice of many of the top Jazz players around. When you try one, you'll see why.

Accessories

Lead

Make sure you buy a good quality lead and learn to "roll" it correctly.
Low quality leads don't last well, spending extra on the lead will ensure it lasts longer.

Clip on tuner

Your bass has to be tuned every time before you play it.
When joining a band on bass, you need to turn up early. You will need to run a power extension lead, get out the school bass amp, setup your chair and music stand. Get out your bass, plug it in and finally with the volume off - tune it with a good clip on tuner.
I personally use and recommend the red SNARK.

Current Street Price: $ 29.95