Sleeping bags should now be rated to EN13537 which is a standard and for each bag will give four ratings: upper comfort, comfort, limit and extreme. Comfort is the rating for the "average*" woman, limit the rating for the "average" man. In practice, choose the "comfort" rating if you are a cold sleeper, "limit" if you are a warm sleeper. The "extreme" rating is basically survival mode and should be ignored for all practical purposes. Some manufacturers/shops split the comfort and limit values to give a sort of average. EN13537 is sufficiently useful that even some American manufacturers use it.
Many (most/all?) synthetic fillings will lose effectiveness over time, say a few years. The fibres in the insulation become brittle and snap so don't trap as much air (which is what actually provides the insulation). Down keeps its effectiveness for much longer - the downside (ho ho!) is that you need to take a bit more care in use and avoid getting it properly wet during use, depending on the shell materials they can cope with small amounts of damp. Down also costs more, especially the better fill ratings like 850+ fill power which also tend to have a better down to feather ratio which also pushes up the price.
Warm, light/compact, cheap: pick two.
In your case with a young child I'd go for zipped synthetic bag rated to the 5-8C range. Synthetic because it will deal with campsite spills better; zipped to give better temperature regulation across a wider range; 5-8C to allow for the degradation of the filling over time. I'd also get a decent compression sack for it so you can get it as small as possible for your pack. The last synthetic bag I bought was a Snugpack but that was twenty years ago, all my recent purchases have been down bags, it's still going but is probably in a sad state. You don't actually state what "cheap" means to you - my idea of cheap might be completely different to yours. Alpkit bags are regarded as decent -
https://www.alpkit.com/products/mountain-ghost-140For late spring to autumn use I prefer a quilt but not everyone gets on with them and they are usually down (there's a few US manufacturers that do synthetic ones). They are lighter for a given temp rating and have a better range of usable temperatures.
*average in this case is a mannequin stuffed with temperature sensing electrodes and dressed in a shell suit