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The Rune Scribe prestige class (from Unearthed Arcana: Prestige Classes and Rune Magic) grants the Living Rune feature at 4th level:

At the end of a long rest, you can choose to increase one ability score of your choice by 2 or increase two ability scores of your choice by 1. At the end of each subsequent long rest, you can alter this choice, reducing the scores you previously increased and improving different scores. (If you chose to increase two abilities, you can opt to alter only one of those choices.)

Since this ability is similar to an Ability Score Improvement, can you pick a feat instead?

That UA Scribe prestige class feature does not mention feats

Granted, this is UA/playtest rather than a finished class or feature, but it's still best to approach the text under the basic assumption of "if it offered that option, it would say so."

Since Living Rune makes no mention of feats, I'd offer that it was not intended to grant a feat change with each long rest.

I'd recommend against, since it only mentions ability scores, but your DM may like to allow a crazy fun option. Here is why I'd recommend against.

Living Rune

At the end of a long rest, you can choose to increase one ability score of your choice by 2 or increase two ability scores of your choice by 1. At the end of each subsequent long rest, you can alter
this choice, reducing the scores you previously increased and improving different scores. (If you chose to increase two abilities, you can opt to alter only one of those choices.)

1. Feats are an optional rule in the PHB. (p. 165) You are already in DM discretion territory as it is.



2. 
   

   Feats are an opportunity cost; you only get to select one when an ASI/feat option arises, and then it's locked in.

   
   
   

   Applying this feature as proposed in the question, the PC could have access to each and every feat in the book, with long rest in between. That's not quite balanced with all other characters needing to make a choice and stick with it.

   
   
   

   Example:
   
   Day 1, you have the Alert feat. You can't be surprised. You then get an idea for where you will be exploring on day 2, so
   
   Day 2, you take the magic initiate feat, Warlock, choosing eldritch blast, hex and mage hand. You then get an idea for where you will be exploring on day 3, so
   
   Day 3 you take Weapon Master and have proficiency in Long bow, Long sword, Rapier, and glaive. You then get an idea for where you will be exploring on day 4, so
   
   Day 4 you take the Lucky feat. You then get an idea for where you will be exploring on day 5, so
   
   Day 5 you take Dungeon Delver ...

   
   

Nobody else has this kind of versatility.

Crazy fun option

Do as you propose, play for a few levels, and then add an answer here about why it was a good idea and why it wasn't. Work with your DM on that. UA is playtest, so why not playtest this and write up a playtest result as an answer? :)

No, you cannot take a feat instead

p. 165 PHB describe the circumstances that allow you to take a feat instead of increasing your ability scores:

At certain levels, your class gives you the Ability Score Improvement feature. [...] [Y]ou can forgo taking that feature to take a feat of your choice instead.

You don't gain the improvements of the living rune ability when you level up, but when you activate it at the end of a rest and it's not the Ability Score Improvement feature, it just happens to have a similar effect.