This english.se answer describes how the word "buff" evolved independently in video game dialect. When was the word "buff" first used in tabletop RPGs to describe bolstering effects, and was it adopted directly from video game culture?
The first occurrence of the term "buff" in a published D&D work I can find is the Psionics Handbook (2001) for 3e, where the astral construct has abilities such as "Buff (Ex): The astral construct gains an extra 5 hit points", with "Improved Buff" and "Extra Buff," meaning the term wasn't being used purely in the sense of "strong."
The Wizards forums start to see the term "buff" used later - 2003 is the first occurrence I can find in the Wizards forums, and at the time it is often enclosed in quotes which indicate it's not in completely common usage. Buff doesn't appear in the 2001 "Common Message Board Terminology" list. Similarly, the first use I can find in the context of "'buff' spells" on the RPG.net forums is September 2002.
The next place I find the term used a lot in a published product, interestingly, is in Living Greyhawk adventures, exploding into wide use in season 4 (~2003-2004) - these were written more informally by community members and you can see the term gain great currency there quickly. Not coincidentally, 2004 is when World of Warcraft launched and became a huge fad.
The next mention I can find is in the 3.5e Warcraft RPG Lands of Conflict (2004) where it talks about someone "buffing their skills" - this could demonstrate the use of the common computer gaming term being imported into D&D in its true form for the first time.
Dungeons and Dragons for Dummies (2005) has an entire section on "Buffing: Making your character and the team better." It also states "The cleric's spells revolve around healing damage and enhancing the abilities of other party members. These enhancements, commonly called buffs, allow the cleric to make everybody around him or her better." D&D For Dummies has a lot of sections employing common CharOp terminology and advice from the time - "buff" is used extensively throughout their discussion of recommended builds, including "the buffer sorcerer" et al. The term is used no less than 52 times in this work so I think it's fair to say it is completely established by this time.
So the term appeared in tabletop land in late 2002 and exploded in use over the next couple years. The term was in use in video games prior to that - a mention from Anarchy Online in 2001 for example, and Everquest players were using it in 1999.
Therefore it seems clear that the term was first imported from normal English (buff, to shine up) into video gaming terminology for enhancements and then found its way into tabletop.
After a considerable amount of wading through links and research I've only found one viable answer. the verb "to buff" is simply derived from the adjective "buff" meaning to describe someone muscular or very athletic, and became used in video games to imply that you're making someone better by making them more buff i.e. "buffing" them.
As to where this came from, It wasn't popularized until video game culture, possibly being first used in MMO's such as WoW for shorthand, and was likely borrowed by Tabletops rather than the other way around
Most video game slang originated from online gaming. often this happens because when needing to type things to friends in the middle of a combat situation, shorthand was alot more efficient compared to trying to type everything out. It's unlikely it would be needed in a situation where you could talk to people out loud and get your ideas across more rapidly and easily, which is why words like "gank" "nerf" "buff" etc became popularized in PVP and large scale games such as MMO's and eventually Mobas.
While I have not found any substantial proof through my search thus far, the logical assumption is that this originated in online gaming and eventually flowed over to tabletop, since many gamers frequent both forms of play.