It does not auto-adjust.

I agree with John Campbell of the giantitp boards on this one. When the Bone Bow says that it

functions as a composite longbow with regard to applying the user’s Strength bonus to damage done with arrows shot from it. (75)

I see no reason to assume that it only applies to parts of the rules text concerning composite (long)bows and their interactions with the wielders Str bonus.

All composite bows are made with a particular strength rating (that is, each requires a minimum Strength modifier to use with proficiency). If your Strength bonus is less than the strength rating of the composite bow, you can't effectively use it, so you take a -2 penalty on attacks with it. The default composite longbow requires a Strength modifier of +0 or higher to use with proficiency. A composite longbow can be made with a high strength rating to take advantage of an above-average Strength score; this feature allows you to add your Strength bonus to damage, up to the maximum bonus indicated for the bow. Each point of Strength bonus granted by the bow adds 100 gp to its cost.

The only clarity issue here is the pricing of the modifier, which differs between the Longbow and the Shortbow

Each point of Strength bonus granted by the bow adds 75 gp to its cost.

Considering that 100 and 75 gp are the base costs for the composite long- and shortbow respectively, it can be assumed that every point of Str modifier costs the full base price for the bone bow as well, which makes the bone bow quite expensive, especially considering the feat requirement.

Should it?

On the one hand, I'd say it shouldn't, because I can't really justify it in character/in world, and my gut says no.

On the other hand, both (most) archers and exotic weapons are well known to be rather weak in D&D 3.5, even compared to other mortals martials. Giving them a boost (which mileage varies widely depending on how often you Str rating changes) is certainly not game-breaking, especially if they need to burn a feat to use it, on top of those other restrictions.

Here's an excerpt from the D&D 5e Basic Rules v0.3:

Hit Points and Hit Dice

At 1st level, your character has 1 Hit Die, and the die type is determined by your class. You start with hit points equal to the highest roll of that die, as indicated in your class description. (You also add your Constitution modifier, which you'll determine in step 32.) This is also your hit point maximum.

So, let's look at that Halfling Rogue! The Rogue has a Hit Die of d8, so at 1st level, they will take the maximum of 8 and add their Constitution modifier, which in your case is +1, for a total of 9 Hit Points.

Beyond 1st level, on level-up you can decide to either roll the Hit Die or take the Average Number listed in the description, which in the case of a d8 is a 5. You add your Constitution modifier to that, too.

The thing you talk about, how a Goblin deals 1d6+2 from a Shortbow presumably, and how the Halfling Rogue has a mere 9 Hit Points. This is a common trouble for many editions of D&D. 1st level characters are fragile, almost always.

I will leave it at this for now, but I will be back after some XP calculation for encounter balancing, as 3 Goblins versus 2 people might be a little tough, which is something you found.

Cool, so it turns out 3 Goblins vs 2 1st level characters is an utterly deadly encounter! Encounter balancing is based on strict experience point calculation in D&D 5e, based on Party Character levels and the experience points a monster would yield. The following information can be found in the DMG on page 82 and the DM Basic Rules v0.3 on page 56:

A 1st level character's experience threshold looks like this:

• Easy : 25
• Medium : 50
• Hard : 75
• Deadly : 100

And when you perform these calculations, you add up the experience thresholds for all of the party members, in your case two 1st level characters, which gives us:

• Easy : 50
• Medium : 100
• Hard : 150
• Deadly : 200

So now we know what kinds of experience point thresholds there are for your party. Now we add up the experience point yield of the 3 Goblins, which are CR 1/4 - 50 XP each, to get 150 XP. This reached the Hard threshold of your party! Unfortunately, it is not that simple, because we must modify that effective difficulty to truly demonstrate the difficulty associated with strength in numbers. In our case, with 3 Goblins, we multiply the effective XP by two, for effectively 300 XP worth of monsters facing your party. This is well over the Deadly threshold of your party!!!

Note: This does not mean you should reward 300 XP for the encounter should the party manage to defeat them all. The XP reward is still actually just 150, to be divided between the players. However, as DM, you can totally just dole out XP if you feel like doing that.

So, as you can see, even two goblins would be a Hard encounter, and a single lone goblin would be an easier one to handle without the imminent threat of character or party death.

I hope I was of some assistance.

SO! Aramis has pointed out that I forgot about the section addressing Party Sizes of less than 3 or greater than 5 characters. Here is the relevant text:

Party Size

The preceding guidelines assume that you have a party consisting of three to five adventurers.

If the party contains fewer than three characters, apply the next highest multiplier on the Encounter Multipliers table. For example, apply a multiplier of 1.5 when the characters fight a single monster, and a multiplier of 5 for groups of fifteen or more monsters.

Which means for us that we don't multiply the Goblin's XP by 2, but rather the next step of 2.5, for an effective 375 XP versus the Deadly Encounter threshold of 200 XP. Two Goblins with a normal un-adjusted total of 100 XP would be multiplied by 2 rather than 1.5, for what is still a deadly encounter at 200 Effective XP.

Relevant Sources

The Player Basic Rules v0.3 and DM Basic Rules v0.3 can be found here on the WoTC site.