First, if you are going to buy your offspring a drumkit make sure you know how much noise they will want to make. If noise is going to be an issue consider electronic drums with headphones, there are even reasonably priced kits that plug into a PC which can get your junior Buddy Rich off to a flying start.
Next you need to think about whether or not this is just a flash-in-the-pan idea (in which case a few pots & pans and couple of chopsticks might fit the bill) or if this could be the start of something big. The temptation with any youngster and any 'new thing' is to get something shiny and nicely presented. Be warned, some of the things advertised here on ebay may look like junior drum kits but they are plastic toys and your kiddies dreams could be dashed due to an unsuitable choice. (I'm probably not allowed to mention specific examples so I will just say they crop up again and again. )This goes for all types of things of course where parents are tempted by the words 'suitable for beginner'. Sometimes this will be OK and means that it is useable but not top quality, most often it means 'I'm describing it like this because an expert wouldn't touch it with a barge pole'. If you aren't a drummer or a musician yourself then ask someone who is for advice. There are bargains to be had both here on ebay and in shops but look carefully.
If you are still in the game at this point then look carefully for something to suit your child. If he or she (yes, girls like drums too) is 3 years old then a brightly coloured plastic toy with flashing lights is probably in order. If they are older then they will want something that is more like a real kit, both in terms of sounds and looks. This is where it gets difficult. Kids of 6 or 7 can be big enough for a 'fusion' kit, a year or two older and they can manage a 'rock' kit. If you aren't familiar with these terms a fusion kit has slightly smaller drums than a rock kit. Both are 'adult' sized so will do the job for many years. Remember kids grow at an alarming rate and will soon grow out of something too small.
Assuming you have gone the accoustic (as opposed to electronic) route there are many perfectly adequate kits out there. Most drums will do the job required of them, sometimes a cheap kit can be hugely improved with some decent new heads (skins). Assuming the drum is round and there is nothing broken, split or damaged then it should function OK. The main problem with cheaper kits is the cymbals. Many cheaper kits have cymbals that are no more than cymbal-shaped disks of cheap brass, an old tea tray with a hole in the middle would do a better job. Whilst cheap cymbals will get your child started they, and you, will soon learn to appreciate the sound of a decent cymbal. Unfortunately good cymbals aren't cheap so maybe that is something that willing grandparents can fund if the kit ever gets more that a few weeks of hammering. Again, there are bargains to be had if you look carefully. Whatever you buy try to make sure it has everything included as it can work out more expensive to buy odd bits and pieces in an attempt to put a kit together....that last elusive piece is always the most expensive. You should look for :
Bass Drum with pedal
Snare drum with stand
Hi-Hat cymbals with stand normally 14 ins diameter
Tom-Tom mounted on bass drum (at least 1...2 if poss)
Floor Tom (side drum) with legs or stand
Crash cymbal 14-16 ins diameter on stand
Ride cymbal 18-22 ins diameter on stand
Stool, also known as a 'throne'
......and don't forget damper or silencer pads for all of the above
My own kit which I use for live gigs cost £67 here on ebay. It is a LeX Hohner which was at the cheap end of budget kits when new. I have changed the heads and bought decent cymbals (Sabian XS20s) but the basic drums are still very much 'budget' ...but they do a great job. I have two Stagg cymbal stands that cost me a fiver each and a stool that was £15. The point is that you don't have to spend a fortune to get something useable. Sound matters more than looks and most people don't get to see the kit anyway as we drummers tend to sit at the back in the dark (I've even sat in a pub fireplace before now when space was tight).
Having paid for something useable how is your child going to use it? The assumption tends to be that drums are easy, all you do is hit them. In fact they are a musical instrument, or set of instruments, in their own right. Without guidance your child could quickly lose interest and we would see their kit advertised on ebay along with the words 'son/daughter lost interest'. That happens all the time with kids but some lessons from a drum tutor might be a good investment to set them on the right track. This is not a cheap option but could be a way to 'try before you buy',many tutors offer a free first trial lesson. If the tutor thinks you child has no co-ordination, no rhythm, no musical sense and no aptitude for learning these things you will be told and you can go look for something else to consume your cash and their interest. If, on the other hand, there is some degree of talent then it can be properly encouraged and nurtured. At the very least a drum tutor can give advice on what to buy or not to buy, they may even know of good secondhand gear going at reasonable prices. Of course if you go the secondhand route then you have to manage the expecations of your child so they are not disappointed by the not-so-shiny-but-fully-functional kit you present them with on Christmas morning.
The above advice really goes for all musical instruments, except maybe a kazoo. Research the market, take advice, look around, don't be tempted by the first shiny thing you see. Consider spending a little more for a proper kit which you can either sell or trade up at a later date.
Hope this helps.