Tunisia in September is still hot - maybe 30+ we went in September with DD when about 9 months and it was too hot for her really, although we managed as only for a day on a cruise. If you fancy Tunisia have a look at Morocco, there are some lovely hotels outside of Marrakech and a great climate. We stayed at a place called Tigmi recently and they couldn't have done enough for the children. Plus there is the exoctic Marrakech nearby for something different from your standard beach trip.
Places we've been with 1yo DD in our travelling days:
Greek Islands - fab, lovely heat and lots of eating choices for her plus beaches and very welcoming to kids everywhere, but 4ish hours flight.
Spain, Italy and Southern France - All lovely, again plenty to eat and do for little ones. Italians very welcoming in restaurants etc, French not so but no worse than here. Beware that in France need a car seat for taxi transfers and in Spain/Italy would recommend as driving a little crazy! Spain was hard work with pushchair, villages/towns hard to negotiate and public transport also difficult.
Only downside to southern Europe is restaurants weren't open when DD wanted to eat - even in big cities. Siesta straight after lunch caught us out if we didn't get fed before 2pm and often evening service won't start until 7pm or later. So worth stocking up at lunch then picking up bread/cheese etc to make her a snack tea and then you can eat out/in later on in evening. Our DD would always very obligingly fall asleep on the walk to the restaurant in her buggy so we had lots of nice relaxing meals out. Not so relaxing now she is nearly 4yo and we have a nearly 2yo DS! They are still pretty well travelled though!
BTW, don't be put off by the flight time to some places, we found DD much easier on flights at 1yo than at 1+, although she is easier again now. They tend to sleep with the motion and fellow passengers are far more tolerant of a baby than an older child and you get plenty of would be entertainers who are missing their children/grandchildren.