I would steer clear of the october half term, ive been there for the last week of the season and i have bad memories of sitting under a canopy clad in a kagool, holding an umbrella playing bingo and still getting soaking wet! The weather at that time of the year is unpredictable to say the least!! (i was in menorca at the weekends either end of the same week flying home inbetween) and the first weekend it was glorious and the second it was diabolical!
Ive had some good deals to menorca for the first week of the summer break 2 years running (around the 23rd july).
Cosmos do some great lates, we booked last year for a s/c AOA for about £180 each and when we got there, we could have taken our apartment in cala blanca, but we were offered the esmerelda in Cuitadella for an extra 140euros between my daughter to upgrade to 4*AI (which we did without a second thought!)
http://www.cosmosholidays.co.uk
The year before we went, 3 adults and my daughter, and got named accom with firstchoice at the flora gardens in cala n bosch for £750 between us. Im sure you will find something!
http://www.firstchoiceholidays.co.uk
Like the previous person quoted, there arent any lowcost airlines that fly into Mahon, apart from the new service with easy jet that has been set up for the summer months flying from both gatwick, and bristol.
Another option is Luton with monarch scheduled, but considering the time of year it would be more worth your while booking a package late as accomodation costs can be high even booking direct - believe me, i love menorca, and have been there 7 or 8 times now all at short notice and this is the best way to book.
you can also look at
http://www.gardenhotels.com i always stay in the garden hotels in cala n bosch, (apart from my defection last year to the esmerelda, but i spent most of my week around the lago garden pool and watching their entertainment!) all are in great locations and service is excellent, and they do sometimes have special deals on for accomodation.
I hope this helps you a little!
I would say hold out for as long as you can, and start checking the websites now for deals for the 23rd july (ish) keep an eye on them, and if needs be put your child through as an adult to avoid under occupancies, as ive noticed that cosmos are charging them this year, whereas they didnt last year on many of their lates.
get in touch if you need any more help!!
Caroline & Kirsten