I too vote for playing Half-Life 1 before playing Black Mesa. I think you’ll enjoy the Black Mesa experience much more, if you’re versed enough in the original to be able to identify all the little nods, improvements, and changes from the original that the devs may have made. I think it’ll be a richer experience being able to witness the huge transition first-hand, rather than playing Black Mesa knowing little more than “this is an improvement of an old 90s game.”
Not only that, but if you play Black Mesa first, you may run the risk of finding it difficult to go back the original, if you decide to try it out later on. I know that for me, at least, it tends to be a lot harder to go back in time with a videogame than it is to go forward. Once you’re exposed to an updated title, it’s just kind of a pain to go back to the original, somehow. Although, you might already be running this risk, since you played through Half-Life 2 already… (Shame on you, by the way! :fffuuu:
)
I also vote for playing the original rather than the source version of Half-Life 1, due to reasons already mentioned, especially by Grey Acumen and coldroll.
And if you’ve got a mind to play through the entire Half-Life 1 Saga, (Half-Life, Opposing Force, Blue Shift, and Decay,) and not just Half-Life 1, then I absolutely second what Spockjedi said, although I would personally recommend playing through the games in a slightly different order, for reasons I’ll mention shortly.
- First, I’d recommend picking up the Half-Life 1 Anthology from Steam. ($15) (If you just want Half-Life 1, both the original and the Source versions are $10.)
- After that, if you’re so inclined, I’d recommend installing HD pack, unless you prefer the original, old-school graphics.
- Lastly, as Spockjedi mentioned, don’t forget about Unlocked for Blue Shift (fixes some bugs that never got addressed when Blue Shift was ported to Steam from GoldSrc from Dreamcast… long story) and also Half-Life: Decay, which was ported by fans from PS2 to PC fairly recently.
The order I’d recommend playing them in is: Half-Life, followed by Decay (with a friend, since it’s meant to be co-op) or Blue Shift. Opposing Force should be saved for last, IMHO. Playing either Decay or Blue Shift after having gone through Opposing Force can be somewhat anticlimactic, I think, for reasons I think you’ll discover. 
tl;dr - Yes, play the original (not source) Half-Life 1 before Black Mesa. Also see bullet points above. :3