Watson, I sewed stuffed animals for 20 years for craft shows and I have two recommendations for you. Sew all of your piecing and/or animal seams with a stitch that has two forward and one backward. I don't know what the official name is, but it is usually included in most machines. This produces a stronger seam for when you stuff the bear. Or, if you do not have that stitch, sew the seams twice-once in each direction. I know that this sounds like a lot of work, but it is easier than unstuffing and repairing a popped seam. It is also more durable in actual use by a child.
The 2nd hint I have will help if you do not have the 3 step stitch or choose to not do double seams, but should always be done with pieced pattern pieces to again protect all those seams when you are stuffing the animal. When you cut out your pattern pieces, lay another layer of fabric down underneath and cut it out at the same time. Use the two layers as one. The inner layer will not stretch like the top does, causing less strain on the seams.
I hope this has helped you, but if you have any questions, please pm me and I will be glad to assist you in any way I can.