It is currently my turn and I am the controller of Ghost Council of Orzhova. My opponent has an untapped Mother of Runes. I cast Mortify targeting Mother of Runes and in response my opponent taps Mother of Runes targeting itself (presumably to give it protection from one of Mortify's colors).
If I activate my Arena in response to the Mother of Runes ability targeting my Ghost Council of Orzhova and my opponent's Mother of Runes
When Arena's ability resolves, the targeted creatures fight each other even if either or both of them were already tapped as the ability started to resolve; if a creature is already tapped, the action of tapping it is ignored (C.R. 101.3, 701.20a; see also C.R. 609.3). A creature targeted by that ability may be tapped or untapped (C.R. 115.1c; see also this thread). (Note that if the opponent you chose controls another creature besides Mother of Runes, they could target that creature rather than Mother of Runes as you activate Arena's ability [C.R. 115.1c, 601.2c, 601.4].)
EDIT (Jun. 11): Correctness edit.
EDIT (Jul. 11): One rule was renumbered with Core Set 2020.
EDIT (Sep. 4): Edit rule citations.
The two creatures fight, and there is nothing to prevent the damage they deal. Presumably Mother of Runes will be marked with lethal damage and so it will die when state-based actions are performed. Then both its own ability and the Mortify will be removed from the stack for not having legal targets when they try to resolve.
EDIT: While you choose Ghost Dad as the Arena ability's target, your opponent chooses a creature they control; this all assumes they can't target anything but the Mother.
Mother of Runes
Ghost Council of Orzhova
Arena
Mortify
It is currently my turn and I am the controller of Ghost Council of Orzhova. My opponent has an untapped Mother of Runes. I cast Mortify targeting Mother of Runes and in response my opponent taps Mother of Runes targeting itself (presumably to give it protection from one of Mortify's colors).
If I activate my Arena in response to the Mother of Runes ability targeting my Ghost Council of Orzhova and my opponent's Mother of Runes
Are Mom and Dad fighting?
EDIT (Jun. 11): Correctness edit.
EDIT (Jul. 11): One rule was renumbered with Core Set 2020.
EDIT (Sep. 4): Edit rule citations.
EDIT: While you choose Ghost Dad as the Arena ability's target, your opponent chooses a creature they control; this all assumes they can't target anything but the Mother.
I̟̥͍̠ͅn̩͉̣͍̬͚ͅ ̬̬͖t̯̹̞̺͖͓̯̤h̘͍̬e͙̯͈̖̼̮ ̭̬f̺̲̲̪i͙͉̟̩̰r̪̝͚͈̝̥͍̝̲s̼̻͇̘̳͔ͅt̲̺̳̗̜̪̙ ̳̺̥̻͚̗ͅm̜̜̟̰͈͓͎͇o̝̖̮̝͇m̯̻̞̼̫̗͓̤e̩̯̬̮̩n͎̱̪̲̹͖t͇̖s̰̮ͅ,̤̲͙̻̭̻̯̹̰ ̖t̫̙̺̯͖͚̯ͅh͙̯̦̳̗̰̟e͖̪͉̼̯ ̪͕g̞̣͔a̗̦t̬̬͓͙̫̖̭̻e̩̻̯ ̜̖̦̖̤̭͙̬t̞̹̥̪͎͉ͅo͕͚͍͇̲͇͓̺ ̭̬͙͈̣̻t͈͍͙͓̫̖͙̩h̪̬̖̙e̗͈ ̗̬̟̞̺̤͉̯ͅa̦̯͚̙̜̮f͉͙̲̣̞̼t̪̤̞̣͚e̲͉̳̥r͇̪̙͚͓l̥̞̞͎̹̯̹ͅi͓̬f̮̥̬̞͈ͅe͎ ̟̩̤̳̠̯̩̯o̮̘̲p̟͚̣̞͉͓e͍̩̣n͔̼͕͚̜e̬̱d̼̘͎̖̹͍̮̠,͖̺̭̱̮ ̣̲͖̬̪̭̥a̪͚n̟̲̝̤̤̞̗d̘̱̗͇̮͕̳͕͔ ͖̞͉͎t̹̙͎h̰̱͉̗e̪̞̱̝̹̩ͅ ̠̱̩̭̦p̯̙e͓o̳͚̰̯̺̱̰͔̘p̬͎̱̣̼̩͇l̗̟̖͚̠e̱͉͔̱̦̬̟̙ ̖͚̪͔̼̦w̺̖̤̱e͖̗̻̦͓̖̘̜r̭̥e͔̹̫̱͕̦̰͕ ̗͔̠p̠̗͍͍̱̳̠r̰͔͎̰o͉̥͓̰͚̥s̟͚̹̱͔̣t͉̙̳̖͖̪̮r̥̘̥͙̹a͉̟̫̟̳̠̟̭t͈̜̰͈͎e̞̣̭̲̬ ͚̗̯̟͙i͍͖̰̘̦͖͉ṇ̮̻̯̦̲̩͍ ̦̮͚̫̤t͉͖̫͕ͅͅh͙̮̻̘̣̮̼e͕̺ ͙l͕̠͎̰̥i̲͓͉̲g̫̳̟͈͇̖h̠̦̖t͓̯͎̗ ̳̪̘̟̙̩̦o̫̲f̙͔̰̙̠ ̹̪̗͇̯t͖̼̼͉͖̬h̹͇̩e͚̖̺̤͉̹͕̪ ͚͓̭̝̺G͎̗̯̩o̫̯̮̟̮̳̘d̜̲͙̠-̩̳̯̲̗̜P̹̘̥͉̝h͍͈̗̖̝ͅa͍̗̮̼̗r̜̖͇̙̺a̭̺͔̞̳͈o̪̣͓̯̬͙̯̰̗h̖̦͈̥̯͔.͇̣̙̝