Oh, to find a Shadowrun game to play in... *sigh*
From what I remember, from a min/max efficiency perspective it's most efficient to either spend your BP to bring an attribute to its racial max value minus one, or else leave it unchanged and instead spend Karma to improve it as soon as possible. Of course, you might not survive too long with abysmal stats, so you may want to throw whatever you can into Edge, as you may be relying on it to save your bacon for a while, at least until you get Karma rolling in.
Here's how it breaks down (I'm presuming Human at this point):
You start at 1 on your stat, which costs nothing.
To raise it up 1 point (up to a value of 2), it costs 10 BP or 5x(new stat value) in Karma, so it costs the same in either Karma or BP.
To raise it up 2 points, it costs 20 BP or (5x2)+(5x3), 25 Karma, meaning it will cost slightly more in Karma than in BP.
To raise it up 3 points, it costs 30 BP or (5x2)+(5x3)+(5x4), 45 Karma, which is 1.5 times what it could cost in BP.
For a score of 5 (because we're adding 4 points to the starting value of 1), it's 40 BP or 70 Karma, or not quite double.
For a score of 6 (racial max), it costs... 25 just for the last point, so a total of 65 BP or 100 Karma, so it's not quite as efficient as having it one below the racial max.
If you're human, the racial max for Edge is 7, so you could get a wonderful 6 points of Edge for only 40 BP, and quite frankly you're going to need all the Edge you can get.
I agree with the suggestion of buying specializations for two karma a piece, from a numbers standpoint it's one of of the better deals you can get.
All of which has to be balanced against the needs of the character of course; you might not be able to achieve optimal efficiency, especially with only 300 BP, but the concept still works for in principle.
Finally, I would add that Complex Forms are VERY efficient BP purchases, costing 1 BP per point of rating each, while it would take a total of 22 karma to purchase and raise one form up to rating 6 vs. 6 BP. But again, you're working with a BP shortage...