I am aware that Mimamsa is about philosophy and means critical reflection.But I am interested in the actual etymology of the word so please stick to that point.
A translation is 'revered thought'. In Chapter 13 of A Critical Survey of Indian Philosophy, Chandradhar Sharma writes (https://archive.org/stream/IndianPhilosophyACriticalSurvey/Indian-Philosophy-A-Critical-Survey_djvu.txt):
The word 'Mimamsa' literally means 'revered thought’ and was originally applied to the interpretation of the Vedic rituals which commanded highest reverence. The word is now used in the sense of any critical investigation. The school of Mimamsa justifies both these meanings by giving us rules according to which the commandments of the Veda are to be interpreted and by giving a philosophical justification for the Vedic ritualism. Just as Sankhya and Yoga, Vaishesika and Nyaya are regarded as allied systems, similarly Mimamsa and Vedanta are also treated as allied systems of thought. Both are based on and both try to interpret the Veda. The earlier portion of the Veda, i.e., the Mantra and the Brahmana portion, is called Karmakanda, while the later portion, i.e., the Upanisads is called Jnanakanda, because the former deals with action, with the rituals and the sacrifices, while the latter deals with the knowledge of reality. Mimamsa deals with the earlier portion of the Veda and is therefore called Purva-Mimamsa and also Karma-Mimamsa, while Vedanta deals with the later portion of the Veda and is therefore called Uttara-Mimamsa and also Jnana- Mlmamsa. The former deals with Dharma and the latter with Brahma and therefore the former is also called Dharma-Mlmamsa, while the latter is also called Brahma-Mimamsa. There has been a long line of pre-Shankarite teachers of Vedanta of whom Mandana Mishra seems to be the last, who have regarded Mimamsa and Vedanta as forming a single system and who have advocated the combination of action and knowledge, known as Karma-Jnana-samuchchaya-vada. According to them, the sutras, beginning with the first sutra of Jaimini and ending with the last sutra of Badarayapa, form one compact shastra. These teachers held that Karma (action) and Upasana (meditation) were absolutely essential to hasten the dawn of true knowledge.