As @Henry pointed out, the Wikipedia list seems to be copied from other lists that have been around on the internet for some time now. A similar one with more instructive explanations (of where the names came from) is found here. As you can see, it is a colourful mix or historically substantiated names and persons from mythology and literature; it also confuses rulers of the city with (alleged, probably mythological) rulers of all of Phoenicia.
Bronze age king names
Specifically on the bronze age section (the names before those given by Josephus):
- Abi-Milku is substantiated through the El-Amarna letters. Please note that Milku simply means king in Phoenician (and similar in other Semitic languages); perhaps the name should rather have been rendered Abi or Abi-King (see below).
- Aribas is borrowed from Homer's Odyssey. See here, page 258, line 3. The name is rendered Arybas in this translation, in other translations (see this German one) it is Aribas (p.190, line 3) as in the list. Obviously the Odyssey, a work of fiction written centuries later and in a far away country, is not an authoritative source on Tyrian bronze age kings.
- Baal-Termeg / Baalat-Remeg is substantiated through the Journal of a Frontier Official (as pointed out by @Henry). This text if written in Hieratic (see Pritchard p.258) which is a short form of ancient Egyptian script and would use lots of abbreviations (this should probably explain the uncertainty of modern scholars on whether the name should be read as Baal-Termeg or Baalat-Remeg; Phoenician script would, I think, render it B'LTRMG anyway).
- Baal: Unclear, but Baal is both the name of the Phoenician main goddess and a generic term for "lord" or "prince". The list mentioned above refers to Pharaoh Merneptah's victory over the Canaanites. This may be the incident recorded on the Merneptah stele in which case the name would perhaps be based on a misinterpretation of a generic "prince" (though I am absolutely not sure about this).
- Pummay / Pygmalion is in the list mentioned above an apparent confusion with the much later king Pygmalion that is linked to the legend of the founding of Carthage, perhaps based on speculation that Carthage was founded much earlier.
Later king names
With regard to the kings in the time after the end of Josephus' list, they are probably a bit better substantiated and (as listed in the annotations of the list mentioned above based on a variety of sources.
A note on ancient names
It is generally hard to identify the exact identity behind let alone the pronunciation of proper names in ancient documents. Take Abi-Milku as an example. We know his name from the El-Amarna letters that were compiled in Akkadian Cuneiform even though this was neither the the native language of the Phoenician author nor that of the Egyptian recipient. Wikipedia lists the full text of letter EA153 in which the name is given (in line 3) as IIa-Bi-LUGAL, LUGAL being the Akkadian Sumerian word for king, indicating the cuneiform character for king, hence presumably read as Melek or Milku (actually MLK) in Phoenician. Or perhaps not, who knows - especially considering that these letters (and a few others) are just about the only written sources we have about Phoenicians of that time.
Note: Pritchard here refers to James B. Pritchard (ed.), Ancient Near Eastern Texts Relating to the Old Testament with Supplement, Princeton Univ. Press, 1969
