The Sankrit root smṛ and the term smṛti predate Buddhism. In the ancient Ṛgveda, smṛ means "to remember" or "to keep in mind." Monier Williams Sanskrit-English Dictionary includes the following for each:
to remember, recollect, bear in mind, call to mind, think of, be mindful of
to remember or think of with sorrow or regret
to hand down memoriter, teach, declare
to recite, to be remembered or recorded or declared (as a law) or mentioned in the smṛti
to be declared or regarded as, to cause to remember or be mindful of or regret
to remind any one of, to wish to remember
f. remembrance, reminiscence, thinking of or upon (loc. or comp.), calling to mind (smṛtim api te na yānti, " they are not even thought of "), memory
memory as one of the vyabhicāri-bhāvas (q.v.)
Memory (personified either as the daughter of dakṣa and wife of aṅgiras or as the daughter of dharma and medhā)
the whole body of sacred tradition or what is remembered by human teachers (in contradistinction to śruti or what is directly heard or revealed to the ṛṣis)
These ancient meanings of "remembrance" and "keeping in mind" were retained in the Buddhist canonical discourses as well as the Theravāda and Sarvāstivāda commentarial traditions. For example, the faculty of sati (satindriya) is defined in SN 48.9 Paṭhamavibhaṅga Sutta as follows:
Katamañca, bhikkhave, satindriyaṃ? Idha, bhikkhave, ariyasāvako satimā hoti paramena satinepakkena samannāgato cirakatampi cirabhāsitampi saritā anussaritā – idaṃ vuccati, bhikkhave, satindriyaṃ.
And what, monks, is the faculty of sati? Here, monks, a noble disciple is satimā, possessing supreme sati and discretion, one who remembers and recollects what was done and said long ago. This is called the faculty of sati.
Likewise, in SN 46.3 Sīlasutta we find the following passage relating to the awakening factor of sati (satisambojjhaṅga):
Yasmiṃ samaye, bhikkhave, bhikkhu tathā vūpakaṭṭho viharanto taṃ dhammaṃ anussarati anuvitakketi, satisambojjhaṅgo tasmiṃ samaye bhikkhuno āraddho hoti; satisambojjhaṅgaṃ tasmiṃ samaye bhikkhu bhāveti; satisambojjhaṅgo tasmiṃ samaye bhikkhuno bhāvanāpāripūriṃ gacchati.
Dwelling thus withdrawn, one recollects the dhamma and thinks it over. Whenever, monks, a monk dwelling thus withdrawn recollects that dhamma and thinks it over, on that occasion the awakening factor of sati is aroused by the monk, on that occasion the monk develops the awakening factor of sati, on that occasion the awakening factor of sati comes to fulfillment through development in the monk.
And SN 45.8 Vibhaṅga Sutta we find the description of right sati (sammāsati):
Katamā ca, bhikkhave, sammāsati? Idha, bhikkhave, bhikkhu kāye kāyānupassī viharati ātāpī sampajāno satimā, vineyya loke abhijjhādomanassaṃ; vedanāsu vedanānupassī viharati ātāpī sampajāno satimā, vineyya loke abhijjhādomanassaṃ; citte cittānupassī viharati ātāpī sampajāno satimā, vineyya loke abhijjhādomanassaṃ; dhammesu dhammānupassī viharati ātāpī sampajāno satimā, vineyya loke abhijjhādomanassaṃ – ayaṃ vuccati, bhikkhave, sammāsati.
And what, monks, is right sati? Here, monks, a monk dwells contemplating the body in the body, ardent, fully aware, satimā, having removed covetousness and displeasure with regard to the world. He dwells contemplating feelings in feelings, ardent, fully aware, satimā, having removed covetousness and displeasure with regard to the world. He dwells contemplating mind in mind, ardent, fully aware, satimā, having removed covetousness and displeasure with regard to the world. He dwells contemplating phenomena in phenomena, ardent, fully aware, satimā, having removed covetousness and displeasure with regard to the world.
It's worth noticing that this description of right sati is qualified by being ardent (ātāpī), fully aware (sampajāna), and satimā. To clarify the meanings of these terms we can turn to the Satipaṭṭhānavibhaṅga of the Pāli Abhidhammapiṭaka, which gives the following word analysis for each:
“Ātāpī” ti. Tattha, katamaṁ ātappaṁ? Yo cetasiko viriyārambho nikkamo parakkamo, uyyāmo vāyāmo ussāho ussoḷhī thāmo dhiti asithilaparakkamatā, anikkhittachandatā anikkhittadhuratā dhurasampaggāho, viriyaṁ Viriyindriyaṁ Viriyabalaṁ Sammāvāyāmo – ayaṁ vuccati “ātappaṁ”. Iminā ātappena upeto hoti samupeto upāgato samupāgato, upapanno samupapanno samannāgato. Tena vuccati “ātāpī” ti.
“Sampajāno” ti. Tattha, katamaṁ sampajaññaṁ? Yā paññā pajānanā vicayo pavicayo dhammavicayo, sallakkhaṇā upalakkhaṇā paccupalakkhaṇā, paṇḍiccaṁ kosallaṁ nepuññaṁ vebhabyā cintā upaparikkhā, bhūrī medhā pariṇāyikā vipassanā sampajaññaṁ patodo, paññā Paññindriyaṁ Paññābalaṁ paññāsatthaṁ, paññāpāsādo paññā-āloko paññā-obhāso paññāpajjoto paññāratanaṁ, amoho dhammavicayo Sammādiṭṭhi – idaṁ vuccati “sampajaññaṁ”. Iminā sampajaññena upeto hoti samupeto upāgato samupāgato, upapanno samupapanno samannāgato. Tena vuccati “sampajāno” ti.
“Satimā” ti. Tattha, katamā sati? Yā sati anussati paṭissati sati saraṇatā, dhāraṇatā apilāpanatā asammussanatā, sati Satindriyaṁ Satibalaṁ Sammāsati – ayaṁ vuccati “sati”. Imāya satiyā upeto hoti samupeto upāgato samupāgato, upapanno samupapanno samannāgato. Tena vuccati “satimā” ti.
“Ardent”. Herein, what is ardour? Whatever mental exercise of effort, exertion, great exertion, enterprise, endeavour, attempt, travail, vigour, courage, exertion that is not lax, not putting aside of (wholesome) desire, not putting aside of responsibility, being taken up with responsibility, effort, the faculty of effort, the strength of effort, right endeavour – this is called “ardour”. With this ardour he is endowed, truly endowed, having attained, truly attained, being possessed, truly possessed, furnished (with it). Because of this “ardent” is said.
“Full awareness”. Herein, what is full awareness? That which is wisdom, knowing, investigation, deep investigation, investigation of (the nature of) things, discernment, discrimination, differentiation, erudition, skilfulness, subtlety, clarification, thoughtfulness, consideration, breadth, intelligence, guidance, insight, full awareness, examination, wisdom, the faculty of wisdom, the strength of wisdom, the sword of wisdom, height of wisdom, light of wisdom, lustre of wisdom, flame of wisdom, treasure of wisdom, non-delusion, investigation of (the nature of) things, right view – this is called “full awareness”. With this full awareness he is endowed, truly endowed, having attained, truly attained, being possessed, truly possessed, furnished (with it). Because of this “full awareness” is said.
“Satimāti”. Herein, what is sati? That which is sati, recollection, recall, sati, remembrance, bearing (in mind), not losing, not confusing, sati, the faculty of sati, the strength of sati, right sati – this is called “sati”. With this sati he is endowed, truly endowed, having attained, truly attained, being possessed, truly possessed, furnished (with it). Because of this “satimā” is said.
And so it's clear that the meaning of sati as "remembrance" and "keeping in mind" was still very much retained in the Vibhaṅga (and in parallel passages in the Dhammasaṅgaṇī).
Likewise, the early meanings of remembrance and keeping in mind are retained in the Milindapañha, which gives the characteristics of sati as "calling to mind" or "noting" (apilāpana) and "taking hold" or "keeping in mind" (upaggaṇhana). These two characteristics are further explained as follows:
“Sati, mahārāja, uppajjamānā kusalākusalasāvajjānavajjahīnappaṇītakaṇhasukkasappaṭibhāgadhamme apilāpeti ‘ime cattāro satipaṭṭhānā, ime cattāro sammappadhānā, ime cattāro iddhipādā, imāni pañcindriyāni, imāni pañca balāni, ime satta bojjhaṅgā, ayaṃ ariyo aṭṭhaṅgiko maggo, ayaṃ samatho, ayaṃ vipassanā, ayaṃ vijjā, ayaṃ vimuttī’ti. Tato yogāvacaro sevitabbe dhamme sevati, asevitabbe dhamme na sevati. Bhajitabbe dhamme bhajati abhajittabbe dhamme na bhajati. Evaṃ kho, mahārāja, apilāpanalakkhaṇā satī”ti...
“Sati, mahārāja, uppajjamānā hitāhitānaṃ dhammānaṃ gatiyo samanveti ‘ime dhammā hitā, ime dhammā ahitā. Ime dhammā upakārā, ime dhammā anupakārā’ti. Tato yogāvacaro ahite dhamme apanudeti, hite dhamme upaggaṇhāti. Anupakāre dhamme apanudeti, upakāre dhamme upaggaṇhāti. Evaṃ kho, mahārāja, upaggaṇhanalakkhaṇā satī”ti.
“As sati springs up in the mind of the recluse, he repeatedly notes the wholesome and unwholesome, blameless and blameworthy, insignificant and important, dark and light qualities and those that resemble them thinking, ‘These are the four foundations of mindfulness, these the four right efforts, these the four bases of success, these the five controlling faculties, these the five moral powers, these the seven factors of enlightenment, these are the eight factors of the noble path, this is serenity, this insight, this vision and this freedom.’ Thus does he cultivate those qualities that are desirable and shun those that should be avoided.”...
“As sati springs up in the mind, he searches out the categories of good qualities and their opposites thinking, ‘Such and such qualities are beneficial and such are harmful’. Thus does he make what is unwholesome in himself disappear and maintain what is good.”viewtopic.php?f=23&t=10600#p161422