Considering how rarely the balestela is mentioned in literature, it’s remarkable that somebody (Leonard Tele of Leigh-on-Sea) in sixteenth century Essex possessed one, called it by that name, and bequeathed it, before anyone had published a book using the word (1570 in Essex Wills IX. 203).
It’s good to see that bequest turning up again in a recent book on timekeeping:
1570 Will of Leonard Tele of Leigh, mariner 20 Apr. in P. Glennie & N. Thrift Shaping the Day: Hist. Timekeeping in Eng. & Wales 1300–1800 (2009) vii. 295 To Robert Camper my ‘balestela’.
Here are the subsequent published mentions:
1574 W. Bourne Regiment for Sea (1592) As for the use and making of the balastela, you shall repare to the book of Martyn Curtes, called the arte of Navigation.
1589 R. Eden tr. M. Cortes Arte of Nauigation [STC 5802 image 78] iii. vi. f. 68, The instrument wherewith the Mariners are accustomed to take the altitudes of the North, they call Balestilia, which is a crosse staffe, whereof we will write heereafter in the nienth Chapter.
But, to come clean, it wasn’t used much because there was a perfectly good English name: the cross staff (OED: cross-staff n. 2a ‘an instrument formerly used for taking the altitude of the sun or a star’, first example 1594). It also had another name, Jacob’s staff (OED Jacob’s staff n. 2a, first example 1559). And a third, arbalest (OED: arbalest n. 3, with only an 1816 reference). And a fourth, fore staff (OED: fore-staff n., first example 1669).
1863 Notes & Queries 8 August 114/1 The Jacob’s staff is the radius astronomicus, the baculus Jacobi, the cross-staff (a name applied in modern time to another instrument), the forestaff, the ballastell, ballastella, or bella stella (Spanish), &c. &c. It consists of a long and a short ruler; the short ruler rides at right angles upon the long one, which perforates the middle of it. If the long ruler he graduated with equal divisions, heights and distances may be obtained by the rule of three: if the long ruler be made to carry a scale of cotangents, angles may be taken directly from the instrument. Both graduations were used.
As to the etymology, Spanish is the most likely immediate source:
1822 Diccionario de la lengua castellana por la academia española (ed 6) 102/3
Ballestilla. Mat. Instrumento de hierro que servia para tomar las alturas de los astros. Instrumentum ferreum arcubalistæ simile astronomis deserviens.
1829 Louis Marie Joseph O’hier de Grandpré Répertoire polyglotte de la marine, à l’usage des navigateurs et des armateurs II. 372
Quart de nonante. s. m. angl. back staff—esp. ballestilla—all. jakobs stab—it. balestriglia—port. balestilhe.
(Back staff above is inaccurate: it was a slightly different instrument, which required turning the back to the sun.)
The word arbalest primarily denotes a crossbow, ultimately from Latin arcus bow and ballista missile-throwing engine. Ballestilla is clearly a derivative of ballista with a diminutive ending.