It might seem that the choice of Basic to use in an Evolution line does not matter that much, as Basics have weak attacks, and their characteristics cease to be relevant once the Evolution card is played onto them. This is not true; there are a few factors that could lead to one particular Basic being favored over others:
Having more HP on a Basic allows it to take more hits, and increases its chances of being able to survive until you evolve it.
Some Pokémon have attacks that damage the Bench, e.g. Dark Tyranitar's Spinning Tail or Arcanine ex's Overrun, both doing 20 damage to Benched Pokémon. A Basic with 50 HP can survive two such attacks while on the Bench.
Contrary to the previous point, sometimes having less HP is desirable too. Swoop! Teleporter can be used to set up self-KOs if you have low-HP Basics in your deck, in order to turn on comeback cards like Pow! Hand Extension.
Suppose you have a 60 HP Basic Pokémon in play with 4 damage counters on it. If you Swoop it away for a 40 HP one, the new Basic will be knocked out. This wouldn't be possible if the Basics in your deck all have 50 HP.
A few evolving Basics have attacks which let you draw cards or search your deck for cards. These would be preferred over similarly-named Basics that only have weak damage-dealing attacks. They can even let you cut down on Supporters — for example, Nidoran ♀ has a Basic-searching attack, so it can get by with Great Ball instead of Holon Mentor for its Basic-searching needs.
Many decks use Jirachi's Wishing Star Poké-Power to set up. Jirachi is not often played as a 4-of, so the way this usually goes is that they will start the game with some other Pokémon in the Active Spot, use a Basic-searching Trainer to find Jirachi, then retreat to Jirachi and use Wishing Star.
If your Basic can prevent a retreat to Jirachi (this includes Paralysis and Sleep), this stops that sequence cold, and they have to draw Jirachi in their opener and start with it to use Wishing Star. If your Basic can inflict a Special Condition on Jirachi, it cannot use Wishing Star until it recovers.
If you play Jirachi yourself, then you should choose Basics that have a Retreat Cost of {C} so that you can retreat to Jirachi while spending only 1 Energy.
A good example is Slugma. There are five Slugmas available, but only one of them has 1 Retreat Cost, and thus it's the one that is chosen every time.
Note that this point only applies if Jirachi is part of your setup. Camler happily plays the Numel with Kindle and 2 RC, as it doesn't use Jirachi, and Kindle fits the deck's game plan. It has Fluffy Berry to assist in retreating, too.
Having Poké-Powers/Bodies on Basics is a liability due to the presence of cards like Cursed Stone and Rayquaza ex δ. Rayquaza ex δ can KO a Benched 50 HP Basic with a Power/Body in one hit with Special Circuit!
Note that some Basic Pokémon (e.g. Feebas, Mudkip) have Poké-Bodies that prevent damage from attacks as long as they are Benched. These are actually worth using, as Rayquaza ex δ cannot snipe them, and Cursed Stone only affects Pokémon with Powers, not Bodies.
Holon Adventurer and Holon's Castform draw more cards if you have δ Pokémon to discard, or on the Bench.
Lickitung δ and Kingdra δ put (more) damage counters on δ Pokémon. They're not very common, though.
You can get 2 copies of the same-named Basic from your deck with Lanette's Net Search if they are of different types. δs help with this.
You should also try to avoid having different Lanette's Net Search targets that share types. This can influence your choice of Holon's Pokémon, e.g. using Holon's Voltorb instead of Holon's Magnemite to avoid clashing with Jirachi.