What is the meaning of 'adhikāra' in Hinduism?
The Sanskrit word अधिकारः (adhikārḥ) has a lot of meanings, but within the context of your question, it simply means one's right, privilege or qualification to perform a certain action. There is a reference to the word in the karmaṇyēvādhikārastē śloka of Bhagavad-gītā (2.47).
Swāmī Gambhīrānanda's tr.:
कर्मण्येवाधिकारस्ते मा फलेषु कदाचन ।
मा कर्मफलहेतुर्भूर्मा ते सङ्गोऽस्त्वकर्मणि ৷৷ 2.47 ৷৷
karmaṇyēvādhikārastē mā phalēṣu kadācana.
mā karmaphalahēturbhūrmā tē saṅgō.stvakarmaṇi ৷৷ 2.47 ৷৷
Your right is for action alone, never for the results. Do not become the agent of the results of action. May you not have any inclination for inaction.
Te, your; adhikarah, right; is karmani eva, for action alone, not for steadfastness in Knowledge. Even there, when you are engaged in action, you have ma kadacana, never, i.e. under no condition whatever; a right phalesu, for the results of action may you not have a hankering for the results of action. Whenever you have a hankering for the fruits of action, you will become the agent of acquiring the results of action. Ma, do not; thus bhuh, become; karma-phalahetuh, the agent of acquiring the results of action. For when one engages in action by being impelled by thirst for the results of action, then he does become the cause for the production of the results of action. Ma, may you not; astu, have; sangah, an inclination; akarmani, for inaction, thinking, 'If the results of work be not desired, what is the need of work which involves pain?'
Who decides if someone has the 'adhikāra' to do or not to do something?
The scriptures. In BG 4.17, Kṛṣṇa mentions karma (action), akarma (inaction) and vikarma (prohibited action):
कर्मणो ह्यपि बोद्धव्यं बोद्धव्यं च विकर्मणः ।
अकर्मणश्च बोद्धव्यं गहना कर्मणो गतिः ৷৷ 4.17 ৷৷
karmaṇō hyapi bōddhavyaṅ bōddhavyaṅ ca vikarmaṇaḥ
akarmaṇaśca bōddhavyaṅ gahanā karmaṇō gatiḥ ৷৷ 4.17 ৷৷
For there is something to be known even about action, and something to be known about prohibited action; and something has to be known about inaction. The true nature of action is inscrutable.
Śaṅkarācārya explains the above as:
Hi, for; there is something boddhavyam, to be known; api, even; karmanah, about action enjoined by the scriptures; and there is certainly something to be known vikarmanah, about prohibited action; so, also, there is something to be known akarmanah, about inaction, about sitting quietly.
Also, Manusmṛti 1.58 states the following:
इदं शास्त्रं तु कृत्वाऽसौ मामेव स्वयमादितः ।
विधिवद् ग्राहयामास मरीच्यादींस्त्वहं मुनीन् ॥ ५८ ॥
idaṃ śāstraṃ tu kṛtvā'sau māmeva svayamāditaḥ |
vidhivad grāhayāmāsa marīcyādīṃstvahaṃ munīn || 58 ||
Having propounded this Law, he himself, first of all, taught it to me with due care; I then taught it to Marīci and other Sages. — (58)
Medhātithi's commentary (manubhāṣya) on the same:
In the present context the term 'Law' stands for the whole collection of Injunctions and Prohibitions contained in the Smṛtis, and not for any particular treatise; as this latter was composed by Manu; that is why the Treatise is called ‘Mānava’ (of Manu); otherwise [ i.e., if the Treatise were the ‘Law’ propounded by the Imperishable One], it would have been ‘Hairaṇyagarbha,’ ‘of Hiraṇyagarbha.’
Or is the meaning very arbitrary i.e., not written down in scriptures?
Most actions (rights & duties) are discussed in various smṛtis and dharma śāstras, but not every rule is written down in scriptures as Kṛṣṇa states in the Mahābhārata:
Many persons say, on the one hand, that the scriptures indicate morality. I do not contradict this. The scriptures, however, do not provide for every case.
But, apparently, everything can be deduced from scriptures:
What should be done and what should not cannot be ascertained easily. Everything can be ascertained by the aid of the scriptures.
So when in doubt, it's best to approach one's Guru or elders to determine the proper course of action.