Hmmmm....well, I've never tried apricots before! I use dried apricots and so those I boil until they are plump. BUT I get the same thing with the skin on peaches and plums. It took me a few tries to figure out how to avoid it, and it seems to be that they are OVERcooked, not undercooked. I think that because apricots are so small, I would put less water in (half an inch) and cook them for not quite so long. Maybe try 15 minutes, stick a fork into it and see if it's soft, if not soft then test in 5 minute increments until you find the right time....but prob no longer than 30 min as they are very tiny.
Also, I find if you let them sit on a plate and cool for 20 minutes or so, the skins are easier to get off. When they are hot, I can't get them off of ANY fruit!
Now I take the lazy way with peaches and plums and just keep the skins on, as they puree up just as well as the fruit itself, but I think that this isn't recommended until your LO is about 8-10 months old. Whereas I figure if it's soft enough to puree and blend right in, then it's soft enough for DS to eat!