Yes..I use grass seeds quite a lot in work..and if you only want to 'add' to the existing lawn it is reasonably simple job.
You will need 'hardwearing lawn' type of seed mixture for small lawn to able to take any amount of wear. Usually these 'hard wearing' mixtures are rye grass based. Rye doesn't produce the prettiest of the lawns and it has tendency to grow quite quickly so you will be mowing often during growing season and you cannot mow the lawn very short..rye is not that type of grass. Because the height of the cut have to be slightly higher and the coarseness of the grass, it will wear well and able to take quite good amount regular wear (which small lawn will get).
There is some amount of preparation involved before using the seed. First of all..you need to wait until the worst of the weather has gone and you can see some new growth on existing grass...that will tell you that the soil temperature is starting to get warm enough for seed germination. You need to give good rake to the existing lawn to remove any moss and other dead 'bits' and roughen the surface for the new seeds. When the weather is right..sprinkle the seeds (I'm sure there instruction on the packet for the quantity and 'how to'), 'walk' the seeds in and to level the surface and hopefully within next 10-14 days most of the seeds have germinated. To give that new surface fighting chance to survive..you mustn't 'use' the lawn yet..the new 'seedlings' have their roots near to surface and won't be able to take any trampling or even cut with mower! Allow the grass to grow quite tall good 4-6" and then take little off from the height ...just little tidying up job with mower and carry on like that for other few weeks. Eventually you see the new grass blades getting thicker and more 'grown up' and that's when the roots have got deeper and the grass is able to start coping with wear. HOW EVER!....your 4 yr old still have to behave with roughest of the games on the surface as it is not fully 'kid proof' for some time yet.
So yes...new lawn from seed is achievable, its cheap, its not difficult but it is slower to mature and you will have to keep 'off' from it for some time to have good success.
