Despite the apparent naturalness for deriving pontifex from pons, pontis, we're not so sure that's the right etymology.
Over the years some have made the case for this derivation, including Judy Hallet back in 1970. As early as the first century BCE, though, this derivation has been contended.
I'll skip over the ancient evidence since Joonas Ilmavirta covers Varro and Scaevola's theory. In more modern times, and with better knowledge of Italic languages, other theories have arisen.
Older Theories Summarized
Roland Kent included a few post-Classical suggestions, including:
pons meaning "path" (cognate with Greek πάτος (patos) and Sanskrit panthan-), where the pontifex would "lead the way in processions" (Isidore included this in his Etymologicum Magnum).
Kuhn's suggestion that pons meant path, but religiously speaking, so a "Pfadbereiter, der zum Pfade der Goetter leitet" (pathmaker, who directs the paths of the gods).
ponti- to the root pū 'purify': hence pontifex 'maker of purifications, of expiatory sacrifices.'
Pfund's suggestion ponti- = Oscan pomtis 'five times': hence pontifex 'the one making the fifth,' that is, 'maker of calculations' for calendar purposes, or 'the Five Actors' in some ceremonial performance.
Kent dismisses these almost out of hand, especially anything to do with 'five' for its irrelevance to Roman religion and the pū and pote- suggestions for their phonetic difficulties.
He concludes by presenting Sanskrit and Avestan evidence for "paths to the gods" and decides on that etymology.
Defense of Tradition
Judy Hallet in her 1970 paper on the topic brings up a good point in that many titles held by special offices once connoted something different than their current status would indicate. For example, she mentions the teichopoioi (Greek for "wall-builders") who "were concerned with financial matters" in later periods. Even the Roman office of quaestor was originally the quaestor paricidii and was tasked, originally, with looking for murderers. Therefore, it is not impossible that indeed the original meaning of pontifex is bridge-builder, and their original task was to build and look after bridges (in particular the Pons Sublicius). She also points to Dionysius of Halicarnassus 1.38.3 where he states that the pontifices still (first century CE) make sacrifices at bridges. The clearest meaning should be upheld.
More Modern Theories
This revival is impeded, however, by Bourdellès, who shows that the second root in pontifex cannot be ancient, that it uses a much more recent suffix. Bourdellès therefore goes back to the "five" root, but finds support in Hittite and Tocharian evidence that "five" can signify all, and further that this was an ancient designation in Umbrian that the Romans then borrowed. Honestly, I don't buy it and not many scholars have either.
Bernard Kavanagh also rebutted Hallett's points, and pointed out that the ancient evidence could have entirely been made up post-Scaevola. There is no good ancient evidence connecting them to bridges outside of a story probably concocted to fit the apparent etymology. He points out that Cicero said that the original number of pontifices at the foundation of Rome was five:
Idemque Pompilius et auspiciis maioribus inventis ad pristinum numerum duo augures et sacris e principum numero pontifices quinque praefect... (Cic. Rep. II, 26).
He then derives pontifex from the Sabellic punt- (n.b.: Sabellic is the Italic group that encompasses both Oscan and Umbrian as well as Sabine, Volscian, and Samnite, but not Latin). The -fex part of their name is connected to facere sacra (to do the sacred rites), which would give them the ultimate meaning of "a member of a board of five who performed (the sacred rites)."
One problem with all theories is that we just don't know what the reality of the early years was. Whether the pontifices were concerned with bridges or metaphorical paths to god or whether they were originally just a group of five, the ancients were known for making up history based on folk etymologies. That said, personally, I think derivation from Sabellic pomti- is the best candidate.
Bourdellès, H. Le 1976. "Nature profonde du pontificat romain Tentative d'une étymologie." Revue de l'histoire des religions 189.1: 53–65.
Hallet, Judith P. 1970. " "Over Troubled Waters": The Meaning of the Title Pontifex." Transactions and Proceedings of the American Philological Association 101: 219–227.
Kavanagh, Bernard J. 2000. "Pontifices, Bridge-Making and Ribezzo Revisited." Glotta 76.1: 59-65
Kent, Roland G. 1913. "The Vedic Path of the Gods and the Roman Pontifex." Classical Philology 8.3: 317–326.
Kuhn, A. 1855. "Pfad, πάτος, πόντος, pons, pontifex." Zeitschrift für vergleichende Sprachforschung auf dem Gebiete des Deutschen, Griechischen und Lateinischen 4: 73–77.
Pfund, Th. G. 1847. Altitalische Rechtsaltertuemer in der römischen Sage, Weimar. (pp. 212ff.)