Is it permissible to obtain a license for mineral investment from a minister of the same tribe, in order to exploit abandoned mountains for personal gain and to help the poor, or is this considered an impermissible favoritism at the expense of the rest of the people?

Whoever assumes a ministerial position or similar role must fulfill the trust and avoid favoritism, as Allah Almighty says: ﴿Indeed, Allah commands you to render trusts to whom they are due﴾, and His saying: ﴿O David, indeed We have made you a successor upon the earth, so judge between the people with truth and do not follow [personal] inclination.﴾

If there is a benefit in extracting and investing in minerals, then this should be accessible to everyone. If there are many interested parties, then the way is to issue fair tenders. "Direct assignment" is a gateway to corruption and injustice, and one must beware of falling into it. The advice is to put projects and tenders out for bidding so that whoever wishes can participate, and the one who offers the best proposal and lowest cost wins.