There are several approaches to this problem, none of them accurate as the human happiness is internal subjective experience and thus not verifiable by people other than the afflicted (it is not intersubjectively verifiable).
Please see below several methodologies for pricing the non-economical values such as happiness developed by economists:
Willingness to pay
This problem can be viewed as ‘what is the market price of an item not sold on the market’. To solve it, economists can look whether happiness cannot be, at least in theory, bought on the market. For example, if a pharmaceutical company wants to test some drug that causes unhappines and has to pay USD 1000 for a person to take the test and willingly be unhappy at some level, then the price of this level of unhappiness is USD 1000.
This can be applied when the lack of happiness is directly connected with measurable expenses, like therapy bills, antidepressant drugs, lost income, etc. In this methodology, if a decrease of a happiness is connected with cost of USD 1000 in some period, its value in this period can be measured at not less than USD 1000.
Comparability with damages won in courts
In this method you must find a statistical sample of court judgments, in which compensation was granted for non-pecuniary damages (e.g. for work related undue stress and dissatisfaction) under similar circumstances. As the circumstances can never be the same in different cases (happiness always is subjective and have different economical and social context for each individual), you rank the judgments (e.g. from the weakest to the strongest undue stress and dissatisfaction) and try to place your case on the X axis against court cases. This should result in a point or a range at which unhappiness at question should be valued.
The methods above are connected with many technical problems and are highly criticized. For example, such issues are usually tackled in economics by analyzing the relationship between the marginal cost and one unit of happiness. But what would be a unit of happiness? I recommend further reading of some legal papers on this issue as well as Steven Kelman criticizm in his paper ‘Cost-Benefit Analysis: An Ethical Critique’, where he argues against putting dolar values on non-market benefits and costs at all.
Hope this helps.