I would have thought that it would be fine considering it is a fruit tree but Googling brought this up...
Mango wood contains a sap that is located at the base of the stem, branches, and trunk. As a result, a recommendation is made never to burn mango wood as it emits a smoke that is full of potent irritants. Plus, Mango trees are highly suseptible to a number of diseases and pest infestations, including the fruit fly, black twig borer, sooty mold, and southern green stink bug to name a few. Pesticide application is generally necessary to maintain the health of these trees. Because of the chemical application, Mango wood does not make for good BBQ! Stick to forest producing products for the safest woods for BBQ!
So I guess no go on the wood, but you could still cut it down for the neighbor if she is an older lady