You cannot ask for the meaning of a single word in a phrasal verb. That's the same as asking for the meaning of "up" in "I came up with a solution". You can't translate just "up", you translate "come up" as a whole. It's the same here.
The phrasal verb you are looking for is "tener que ver", defined in RAE's dictionary as follows:
tener que ver alguien o algo con otra persona o cosa
- loc. verb. Haber entre ellos alguna conexión, relación o semejanza.
Translated: "To exist a connection, relation or likeness between two people or two things". That's the same as the English expression "to have something to do", as you properly note in the question.
Nonetheless, maybe a better question would be why do we use the verb "ver" in that phrasal verb? Well, we can consider that the sight is the most important sense we have to be aware of our surroundings. So through the sight is how we better perceive the likeness between two similar objects. This is acknowledged in a couple of meanings of the verb "ver":
prnl. Traslucirse la imagen o semejanza de algo. El carácter del padre se ve en el hijo.
prnl. Dicho de una cosa: Parecer o evidenciarse. Se ve que tendremos elecciones pronto.
Those two examples can be translated as "the personality of the father is perceived in the son" (you perceive the likeness of the personalities) and "it seems that we're going to vote soon" (you perceive something that is likely to happen).
The phrasal verb "tener que ver" is as old as the Spanish language itself. Here you have an example of its usage in the XIII century, in an old form of "tuviere que ver":
Si el mançebo asoldadado ouiere que ver con la clauera de su sennor & lo pudiere prouar con testigos, el mançebo pierda la soldada que ouiere seruido & saquenlo de casa sin calonna njnguna.
Anonymous, "Fuero de Cuenca", 1284-1295 (Spain).
An attempt to translate that:
If the hired youngster has something to do with the keeper of the keys of his master's treasure, and that can be proven by witnesses, may he lose his wage and be thrown away from the house without further judgement.