French <bidet> 'small horse, pony, nag' (attested since 1564); 'basin for washing private parts' (since 1739); 'trestle' is of obscure origin. It appears to be connected with Old French <bider> 'to trot' (15th century) and <rabider> 'to run toward quickly' (14th c.). Italian <bidetto> 'small horse' (since 1598) is evidently borrowed from French, not the other way round. Savoyard <bidet> means 'donkey'.
Diez hypothesized a Celtic root *bid- 'small' attested nowhere else. Gamillscheg suggested a phonologically unjustified connection with the root of Piemontese <bodero> 'thick' and related words, an obscure group in itself (Meyer-Lübke, REW 1085, 1182a). Bolelli regarded the verb <bider> as primary, and noting its isolation in the FEW, ascribed it to onomatopoeia, but without citing any allegedly onomatopoeic formations in *b-d- elsewhere to support this (L'Italia Dialettale 17:155-6, 1941).
Trying to do better, we look first at French <vider> 'to empty, leave, vacate' as a phonological parallel, along with the corresponding adjective <vide> 'empty, void, vacant'. The former continues OFr <voidier> ca. 1160, <vuidier> 1313; the latter <voide> ca. 1100, <vuide> (fem.) ca. 1261, <vuit>, <vit> (masc.) ca. 1200. Early etymologists attempted to derive these from Latin <vidua:re> 'to deprive of', <viduus> 'deprived, bereft, widowed', but this requires ad-hoc assumptions. The acceptable solution was given almost simultaneously by Schuchardt (Romania 4:256-7, 1875) and Thomsen (ib. 257-62). These words are to be referred to Low Latin *vocita:re 'to empty' and *vocitus 'empty'.
In this position it appears that Latin [k] first underwent affrication to [c^], then voicing to [j^] (falling together with the reflex of Lat. [g]), then lenition to the semivowel [j], resulting in *-oji- which was reduced to the diphthong *-oi- in Continental West Romance (e.g. Old Italian <voito> 'empty, void', later <voto>, <vuoto>; <voitare> 'to void'). It is noteworthy that Lat. <co:gita:re> 'to think' has yielded OFr <cuidier>, OIt <coitare>, later <cotare>, Old Spanish <coidar>, later <cuidar>. Not only has the k/g voicing distinction been neutralized in this environment, but also the o/o: length distinction.
It must also be remarked that OFr <cuit> 'cooked' (continuing Lat. <coctum>) retains the diphthong today, but <vuit> and <vuide> became <vit> and <vide> (at different times in various dialects). This is to be explained by the dissimilative effect of the preceding labial. It might be objected that OFr <puiser> 'to draw liquid' (since ca. 1160) has not been similarly monophthongized by the labial. This verb cannot be separated from its primary noun <puits> 'well' (<puz> ca. 1120, <puiz> ca. 1165, continuing Lat. <puteum>), whose vocalism has been influenced by Germanic words for 'pit'. Frankish *putti can be restored on the basis of Old High German <pfuzzi> (cf. Old English <pytt>).
This suggests that OFr <bider> may have arisen by labial dissimilation from *buidier, earlier *boidier, continuing a Low Latin *bocita:re (*bog-, *bo:c-, *bo:g-). No source for any of the four possible roots immediately shows itself. It may be, however, that the simple verb was extracted from a prefixed verb which underwent regular changes. Pack animals carry packs on both sides, so a very plausible base for the verb could be Celtic *ambi-wogos 'carrying on both sides'. The prefix *amb(i)- is well attested in Gaulish, and reflexes of PIE *weg^H- 'to bear, carry, transport' are known in other Celtic languages. Formation of the active /o/-grade *ambi-wogos would be parallel to Greek <amphíxoos> 'polishing on both sides'.
Difficulties crop up in attempting to derive a Gallo-Latin *ambogita:re (whence *bogita:re and <bider>) from Gaulish *ambivogos. Haplology has evidently produced *ambosta: from Celtic *ambi-bosta: 'that which one gets both hands around', independently in Insular Celtic (Old Irish <imbas>) and on both sides of the Pyrenees (OSp <ambuesta>, Piem. <ambosta>, etc.; REW 411b; Bolelli, ItDial 17:139, 1941). To develop in place from *ambivogos, Gallo-Latin *ambogita:re would require that *ambiv- be either progressively haplologized to *amb-, or assimilated to *ambib- before the haplology. Neither seems likely. In fact if Anvers (Flemish Antwerpen) continues the name of the ancient Ambivarii, while Ambières continues that of the Ambibarii, the reflexes of *ambiv- and *ambib- have remained distinct into modern times. On this basis Gaul. *ambivogos might be expected to yield a Low Latin *anvogita:re, early OFr *anvoidier, which cannot produce the /b/ of <bider>.
The dialect of Perpignan has <abit> 'grandfather', against Paduan <aví>, from Latin <avi:tus> 'grandfatherlike' (Tappolet, Die romanischen Verwandtschaftsnamen 63, 1895; REW 834). This suggests early fortition of intervocalic -v- to -b- in (at least some) northern Pyrenaean dialects of Late Latin, so that a Gaulish loanword *ambivogos could indeed become *ambibogus, and perhaps by early haplology *ambogus, here. Such a word, if it found its way into central Gaul, could then yield Gallo-Latin *am(bi)boga:re 'to carry (packs) on both sides' with a frequentative *am(bi)bogita:re 'to carry (packs) on both sides repeatedly or regularly, to be a pack-animal'. Now, we have <bu:rere> 'to burn' already in the Latin Dioscorides by redivision of <amb-u:rere> 'to burn around', and a by-form *bu:ra:re implied by the gloss "buratum : incensum" (CGL 5:272.43). Thus nothing obstructs the extraction of Low Latin *bogita:re 'to be a pack animal, to move like a pack animal, to trot' which will give us OFr *boidier, *buidier, <bider>. The noun <bidet> stands to <bider> as <jouet> 'plaything' to <jouer> 'to play' and other deverbatives in -et to their verbs.
To explain the presumed route of *ambivogos through the Pyrenees, one possibility is that Asturian ponies were exported under this name as pack-animals. A heavy ancient Celtic presence in Asturias is undeniable and to me it is reasonable that this was the P-Celtic homeland. The Asturian pony was known to Pliny (8:166) as <celdo:>, which has a variant <thieldo:>, apparently related to Basque <zaldi> 'horse' and Berber <aserdun> 'mule'. I do not know whether this word is native to the area or borrowed from Phoenician (as I suspect the 'silver' word is), but its use does not preclude a separate Gaulish term for the animal in its capacity as a beast of burden.
By way of disclaimer, I must admit that I do not KNOW whether the foregoing scenario is historically correct, or even close. It is merely an outline of ONE possible mechanism by which <bidet> MIGHT be explained without recourse to onomatopoeia or an isolated root.
DGK