Alternative to lead-acid battery on power wheel toy
Having an amp rating HIGHER than the original battery is ideal. It will only try to deliver the car the current it needs. If the toy asks for more than the battery can offer, then you have a problem.
However, you need to design the battery bank to safely fail when a cell dies, otherwise you can cause a cascade failure of venting and possibly igniting cells. A very bad thing.
Here is an example of what I would do:

If we are using Panasonic 18650 3200mAh Li-ion batteries, then we know their specs are 3.6 nominal voltage and 10A max continuous current draw.
First, calculate the bank characteristics. 10.8 / 3.6v ~= 3. We will need to use 3 batteries in series to pull the voltage up to spec. Max charge should be around 12.6v, likely nearly identical to the original battery.
Assuming the old battery had a 30 amp fuse, we would need a minimum of 3 rows of 3 batteries to get the bank to the minimum requirements at nominal voltages. However, I would add 2 extra row for overhead. This would total 15 18650 batteries, 5 sets(in parrellel) of 3 batteries(stacked/in series).
When charging the cells, you should charge them in parrellel. 
What I would do is make a separate set of terminals on the battery that connect to all of the batteries in parrellel. from these terminals, charge the battery at no more than 1 amp per cell at no more than 4.2 volts(for 18650 batteries, you need to find the nominal charge rate of the batteries you plan to use if you go another route). Charging batteries in parrellel will naturally balance, as long as those batteries are new batteries married to each other at birth. I have never trusted series charging because of all that can go wrong. 
This would give you a total of 5*3200 mAh for a total of 16000mAh and 173 Wh. 
This is simply an example to get you familiar with the calculations. A battery bank such as this would easily set you back USD $100.
It is important to ensure each cell has overcurrent protection of an amperage lower than the battery is rated for(roughly 2 times the C rating).  Without this, when a cell dies, it can cause an overcurrent cascade, and this is a very very bad thing.